I agree with most of what you said, but to say that modern Finns and Hungarians are descended from older Uralic peoples is not something I think is accurate. Their culture descends from them, yes, but there was probably a lot of assimilation of local peoples when the Magyars and etc. migrated to their final destinations; we shouldn't confuse cultural heritage with biological ancestry.
Oh yes, I know that. I was trying to give a simplified explanation, I didn't want to go into the whole genetics vs. culture thing, but yes of course the Magyars probably picked up people from all sorts of other cultures when they lived on the steppe (Iranic and Turkic peoples certainly) and they then merged with the Slavic people (and Avars) who were already in Hungary before they arrived. But that's pretty much the same thing that happened to all peoples in Eurasia.
We now know from Genetics that most of the time cultures and languages spread by assimilation, rather then outright replacement of previous populations. I'm quite sure that the modern Hungarians are probably significantly genetically descended from the people who had been living on that land since the Stone Age (who were presumably neither Indo-Europeans, nor Uralics).
In Ireland there has been plenty of genetic testing done over the last decade and it shows that we are mostly descended from the first Mesolithic colonists (like probably 80% to 90%), with little bits of input from the various groups who arrived later (Neolithic people, Beaker People, Celts, Vikings, English, etc.). So although Ireland became culturally and linguistically Celtic (and thus Indo-European), we are mostly not descended (genetically) from Celtic peoples of the Alps, or from Proto-Indo-Europeans of the Ukrainian steppes. But there is still a clear historical connection to those cultures that can be established.