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I have not defined what a Traditonal Catholic is, but if I were to do this, I would refer to this motto:

We are what you once were.
We believe what you once believed.
We worship as you once worshipped.
If you were right then, we are right now.
If we are wrong now, you were wrong then.

The only reason why the word traditional appears before Catholic is because as the authors of this letter say:"never before in the Church’s history has it become necessary to coin a special term to describe Catholics whose practice of the Faith has simply remained unchanged in the midst of a totally unprecedented liturgical, pastoral and even theological upheaval."

I believe and practice the same faith my mother did, the same faith my grandmother did, and the same faith my great grandmother did.

If a conciliar Catholic presented himself for Holy Communion at a trad chapel, would he be allowed to receive? If yes, WHY?

I don't understand this question, the differences are in interior belief and spiritual practice, if a concilliar Catholic, presented himself to receive Holy Communion in the prescribed respectful way at my chapel, how would the priest know his interior disposition, barring extraordinary charisms like those of Padre Pio?

I guarantee you any known public heretic, like those who support abortion publicly, like rainbow sash wearers, etc. would be denied Holy Communion, no ifs ands or buts. And I also guarantee you that my priest would give his life before he allowed the Body of Our Lord to be profaned in front of him.