Hi Incredulous. I agree the traditional Mass unites us. It gives us Unity of Worship, which Unites us in Faith, because Lex Orandi, Lex Credendi. There are, as we know, Two Bonds in the Mystical Body of Christ that unite all the faithful, the Bond of Faith and the Communion of Charity. We are united because we believe in and love Jesus and Mary, and because we love one another in Christ's Body. But there are and have always been wounds to unity, like apostasy, heresy and schism, by which the Unity of the Church is attacked by those who should be Her members. Among traditional Catholics, we should love God and love each other very strongly, to keep the Bonds of Unity strong, and so that we may all grow in Grace day by day, both individual members and the whole Church. 

Regarding the New Mass, I will cite an article on some of Archbishop Lefebvre's statements about it: 

From: https://sspx.org/en/what-archbishop-lefebvre-said-about-new-mass

"It is true that prudence might suggest to this or that priest “not to refuse the new Ordo for fear of scandalizing the faithful” by their witnessing his apparent disobedience to the bishop.[40] Such a priest should, however, “keep the Roman Canon which is still permitted, and say the words of consecration in a low voice according to the old form, which is still allowed.”[41] When Archbishop Lefebvre was absent on a Sunday, the seminarians would go and assist at Mass together at the Bernadine convent of La Maigrauge where an old monk celebrated the New Mass in Latin. The archbishop was not a man to rush souls. He allowed himself time to seethe fruits more clearly in order to pass better judgment on the tree. He also wanted to hear the opinions of his colleagues in the episcopate, and find a consensus among his friends ...

At the time, Archbishop Lefebvre’s position was not quite as categorical. He considered that the New Mass was not heretical, but as Cardinal Ottaviani had said, it represents serious dangers; thus in the course of time, “Protestant ideas concerning the Supper would be unconsciously accepted by the Catholics.” This was why children had to be taught the fundamental notions about the Mass. However, “it is an exaggeration to say that most of these Masses are invalid.” One should not hesitate to go a little further to have Mass according to the Roman Ordo; but “if one does not have the choice and if the priest celebrating Mass according to the Novus Ordo is faithful and worthy, one should not abstain from going to Mass.”[44] ...
The problem of assisting at the New Mass

Some priests were torn between the need to keep the Faith as expressed by the traditional Mass and a desire to be obedient as they saw it. In the early days of the reforms, Archbishop Lefebvre advised them to keep at least the traditional Offertory and Canon and to say them in Latin. His advice to the seminarians as to the faithful was remarkably moderate in tone for one who was first to step up to the breach to repel the New Mass.
He exhorted them:

Make every effort to have the Mass of St. Pius V, but if it is impossible to find one within forty kilometers and if there is a pious priest who says the New Mass in as traditional a way as possible, it is good for you to assist at it to fulfill your Sunday obligation."
 

One can counter the dangers for the Faith through solid catechism:

Should all the world’s churches be emptied? I do not feel brave enough to say such a thing. I don’t want to encourage atheism."[10]
 

... [please see the link for more]

These statements are incompatible with the view that the NOM never obtains any graces, or Communion there distributes no graces.

God Bless.  

"It is true that prudence might suggest to this or that priest “not to refuse the new Ordo for fear of scandalizing the faithful” by their witnessing his apparent disobedience to the bishop.[40] Such a priest should, however, “keep the Roman Canon which is still permitted, and say the words of consecration in a low voice according to the old form, which is still allowed.”[41] When Archbishop Lefebvre was absent on a Sunday, the seminarians would go and assist at Mass together at the Bernadine convent of La Maigrauge where an old monk celebrated the New Mass in Latin.

Xavier, you may have already read the rebuttal/explanation for the above quote (and also the FULL quote) from Sean's thread a few weeks ago.  Here is a summary of what and why +ABL said above:
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1) He was speaking of the time of 1980ish, when many of the novus ordo priests WERE UNQUESTIONABLY PRIESTS, having been ordained in the old rite.
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2) He was telling such valid, novus ordo priests to say mass using the TRADITIONAL canon and using the OLD FORM.  The quote keeps going, wherein +ABL told such priests to also use the TRADITIONAL offertory prayers and the TRADITIONAL communion (i.e. not communion in the hand).  In other words, +ABL was telling novus ordo priests to say a TLM mass.
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3) It follows then, according to the above qualifications, that in the early 1980s, it was possibly allowed to attend such a mass, as it was said by a valid priest and was as close to traditional as possible, and was essentially a TLM.
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At the time, Archbishop Lefebvre’s position was not quite as categorical. He considered that the New Mass was not heretical,

When the new mass is said using the TRADITIONAL prayers of the offertory, canon and communion, yes, it's not heretical.
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as Cardinal Ottaviani had said, it represents serious dangers

This is a gross underestimation and falsification of what +Ottaviani said.  He condemned the new mass, in its purest/theoretical form.
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However, “it is an exaggeration to say that most of these Masses are invalid.”

At the time...in the early 1980s...when priests were valid and if using the TRADITIONAL prayers, then this is why +ABL said the above.
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“if one does not have the choice and if the priest celebrating Mass according to the Novus Ordo is faithful and worthy, one should not abstain from going to Mass.”

Faithful priest = one who says mass using the TRADITIONAL prayers (i.e. is faithful to Tradition/doctrine).
Worthy priest = one is unquestionably valid, having been ordained in the pre-V2 rite, by a pre-V2 bishop.
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These statements are incompatible with the view that the NOM never obtains any graces, or Communion there distributes no graces.

You are trying to apply early 1980s validity and traditional use of liturgical prayers to now, over 50 years later, when the % of valid priests who say the new mass are next to 0% and the use of the TRADITIONAL prayers in a novus ordo mass is also 0%.  Your comparisons of these 2 eras as similar is both dishonest and illogical.

I have no idea what +Williamson’s views are.  Being that he’s a little slippery on the new mass, he may also be slippery on the new ordinations.  In my view, the resistance is so scared of being labeled sedevacant, that their knee-jerk reaction is to disagree with many of the sede theological lines-in-the-sand (even those which have nothing to do with the papacy).  I hope I’m wrong but I don’t know. 

Even if he’s not a bishop, he could be a valid pope-elect (theoretically).  Any unmarried, catholic male over a certain age, can be validly elected pope.  You don’t have to be a cleric at all. 

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Quick question. Does the Resistance have an official position -- either explicitly stated or at least existing de facto -- on the new rites of ordination and consecration? Do they require that any priest who works for them be ordained in the old rite, and come from an unbroken line of bishops ordained and consecrate in the old rite?
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And do they tell the faithful to only receive the sacraments from such priests?

No:
Because there is no institutionally monolithic Resistance, there is no official Resistance position on anything.

As far as I can tell "The Resistance" is more of an idea than any kind of concrete organization.

Initially (ie., prior to +Williamson’s entrance into the Resistance), it was not so.  The plan was to prepare for an “SSPX 2B” to replace the old one if it sold out.  But the bishop’s idea of “independent pockets” of Resistance made this situation inevitable.

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Quick question. Does the Resistance have an official position -- either explicitly stated or at least existing de facto -- on the new rites of ordination and consecration? Do they require that any priest who works for them be ordained in the old rite, and come from an unbroken line of bishops ordained and consecrate in the old rite?
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And do they tell the faithful to only receive the sacraments from such priests?

Ok, since my fellow ex-seminarians utterly failed to answer this question for you, I will:

DE FACTO the Resistance does have a position on this. Priests from the Novus Ordo are to be conditionally re-ordained. And the Resistance only makes use of known good or "certainly valid" bishops -- such as +Lefebvre-line.

All priests working in the Resistance are either conditionally ordained by a +Lefebvre line bishop, or they were already ordained by one in the SSPX.
Seems pretty straightforward to me. Classic SSPX position, once again.