I shall be honest I don't really know the Catholic church and I certain don't know Pelosi. But isn't all who receiving communion a sinner? Or has it more to do because the churches position towards abortion?
That is a great question to ask because of course the answer is that we are all sinners. In fact, because we are sinners is why we receive the Eucharist -- we need Christ. However, the Church recognizes venial sin from mortal sin. Mortal sin could be fornication (adultery/masturbation/etc.), worship toward something or someone else, and of course abortion (whether it be committing it, performing it, or promoting it).
The Catholic teaching is that anyone who has committed any kind of mortal sin must first go to confession before receiving communion. He has to reunite with God through reconciliation before uniting with Him through the Eucharist.
However, with Catholics like politicians, they are held to a standard of promoting the faith and having integrity (just like anyone), but because they initiate public access, if they do something publicly against the faith, they have to publicly re-address. Clergy are also held to this standard. This is why it's easier for a politician or some sort of public figure to be excommunicated than someone who is not in the public sphere.
Is the bread being Christ the core of Catholicism? Can't you being a catholic without believing in the catholic Eucharist?
Maybe this is a sidestep from the OP, but I find it very interesting because as a protestant we have so much differences but our centre, Jesus Christ, is the same.
Technically, the answer to that is no. There are tons of Catholics in a theological sense (meaning they have been baptized and confirmed into the Church), but they themselves don't practice the faith, so they aren't Catholic in a faithful sense. Dogmatically speaking, the Eucharist is considered to be the Blessed Sacrament because all of the other sacraments surround this one. This is the one sacrament that actually is receiving Christ literally. So this one sacrament, I'd argue, is especially the one any given Catholic must accept as true. If they don't, all the others fall apart and they've just disbanded dogmatic belief.
You're right that Catholics and Protestants have much more in common than differ. This is why one of the things I take joy in is talking with my Protestant friends, and also trying to get my fellow-Catholics to have involvement with since so many of them misunderstand Protestants, just as so many Protestants misunderstand Catholics. I was a Protestant growing up so I do my best to recognize and understand where they come from. As a Catholic today, I hope more Catholics would take a note from Protestants and fall in love with Scripture -- we were always meant to.