A lot depends upon what the individual in question has done to demonstrate actually leaving the Catholic Church.  

For most people, accepting a heresy on the authority of the Conciliar sect does not necessarily mean that they've departed from the Church and, upon learning the truth and repenting of any sins committed as a result of incorrectly understanding Catholic doctrine/dogma, they reform their lives.  Their intent was to be Catholic according to what the Catholic Church has always taught--they just didn't know what has always been taught.  Thus, the person "we all know" that you describe has probably not really left the Church even though he believes heresy because his intent is to believe the Truths taught by the Church and to be a Catholic.  For him, it is necessary to go to confession to a true priest and, if there is anything else needed, the priest will assist the individual in accomplishing that.

I will note that the person who, in ignorance, believes in a heresy will not be judged for believing a heresy; he will be judged for the sins that he commits that flow from the heresy.  All heresy inevitably leads to other sins.  However, I should add that many truths end up being rejected by such people because they find the heresy is much more comfortable than the truth.  Precisely when the issue of ignorance is no longer an excuse is something only God can know.  However, even if the ignorance is real, we must not participate in the heretic's delusion or, as St. John says, to even say Godspeed to him lest we likewise fall.

On the other hand, when we speak of bishops and popes who have actually studied the faith, who have read the encyclicals and theology manuals, who know that what they are teaching is a departure from the traditional doctrines/dogmas, I believe that they would have to formally and publicly renounce their prior teachings in order to be considered to have returned to the Church.

Quote from: TKGS

when we speak of bishops and popes who have actually studied the faith, who have read the encyclicals and theology manuals, who know that what they are teaching is a departure from the traditional doctrines/dogmas, I believe that they would have to formally and publicly renounce their prior teachings in order to be considered to have returned to the Church.

Thanks. Does anyone happen to know if there's an official teaching, or precedent for this?

You ask for a precedent for the unprecedented situation in which we find ourselves today.  Please note that the above is my personal opinion, but I think it reasonable.

Can you show me a precedent in which the Church has fully re-admitted a person who was formally excommunicated for publicly spreading heresy (as opposed to violating a particular law) and then fully re-admitted to the Church and be allowed to publicly teach doctrine without formally and publicly renouncing his prior heresy?

I know of no time the Church has done so.  People may have been privately reconciled to the Church but never publicly preach, but, being privately reconciled, we would never know about it so he would never be held up as an example to emulate.

So if you're asking if, for example, Bergoglio would simply start teaching the Catholic Faith, would that fact alone return him to the Catholic Church?  If he started teaching the Catholic Faith while refusing to correct his prior errors, the confusion on the part of the laity (and even among the clergy) would be even worse that it is now!  Sewing even greater confusion could hardly be considered repentance.

Reconciliation with the Church of a fallen away Catholic usually begins simply by going to confession. Even if a higher authority is needed (for example, to lift a penalty of excommunication), the confessor himself should say so before absolving. The confessor will then obtain the remission of the penalty from the bishop and communicate it to the penitent. The person is then free to make a good confession and be fully reconciled with the Church.

 :dancing-banana:
Excellent Advice!   Same as given me by Bishop Williamson.  This is exactly what I did after 25 year's wandering in the wilderness of Protestantism.