BRU is fine for a connection. MUC is probably my favorite, as others have mentioned. For BUD, I've always taken the UA flight ORD-MUC, then LH from MUC-BUD.

For someone who is nervous about connection time, I wouldn't recommend FRA. It's a very nice airport, but one of the most unpredictable in terms of the connection time, in my opinion. I've had a 70 minute connection from the US like you mention where I've made it to the gate just as boarding was ending...closer than I prefer after a 9 hour flight from ORD.

You mentioned that flying overall causes you some stress. You'll get lots of advice here about what is possible and "best", but realize a lot of it comes from those of us who do it regularly. I'd suggest, to ease the stress, find a flight with a single connection (in Europe), and with a connection time of 2+ hours. That way if you arrive a little late, you aren't stressing as you deplane. 2+ hours is comfortable for any airport, but even better if you can avoid FRA.

Edit to add: You asked about United. It's fine, they'll get you there. It's not luxury, but no US airline in economy will be. I'd go with whatever gets you the best itinerary and price.