On the Universal Ordinary Magisterium:

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All those things are to be believed with divine and Catholic faith which are contained in the Word of God, written or handed down, and are proposed by the Church either by a solemn judgment or by her ordinary and universal magisterium to be believed as divinely revealed.
(Dogmatic constitution Dei Filius, chapter 3, “Concerning Faith”, Denzinger 1792)

Common sense comes into play here.  Can the pope add an 8th sacrament?  Can the pope dispense with the Sunday obligation?  Obviously not.  How do we know this?  Because we know the basic rules of our Faith.

He changed the words of consecration!  He has no power to do that.  He gutted/edited the Offertory and Canon.  Again, he crossed the line.  Common sense.
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Everyone in the 60s knew it was a bold, daring change.  I don’t understand how 40-50 years later, people argue that a pope “has the power” to change the mass essentially.  

That's circular logic.
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This is only accurate if you qualify the statement and say that the "new rite" is essentially the same as the previous rites.  No pope has the power to introduce a brand new, substantially different from Apostolic times, rite.
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The Pater is not part of the canon, so he didn't change the canon.
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The parameters of Tradition and non-essential changes are VERY BIG parameters.  So when you say a pope can do "anything he desires" that's quasi-contradictory.  It would be like taking your car to the paint shop and saying, "You can paint anything you want, in any style, but I only want my bumper painted red to match the rest of my car."  It's a stupid statement because you say "anything" but then define this word to mean something very specific.
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Again, circular logic.
1.  A pope has no authority to change the essential parts of the mass.  
2.  Paul VI changed the essential parts of the mass.
3.  Therefore, such changes are illegal and without authority.
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Inserting the idea that he's illegitimate, AS A POPE, makes no sense, because it's his act that's illegitimate, not his papacy.
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If you want to argue that he's not the pope based on 10,000 other reasons, then ok.  But simply because a pope issued an illegal order doesn't mean he loses his entire authority.