Burgundians seem to be strictly castle knight civ, as they don't have bloodlines.
It depends. In practice, paladin is rather rare sight, even if you play Franks or Lithuanians, so the deal with Burgundians is that the upgrade comes up earlier and is more affordable and without bloodlines it's still better than cavalier with it. So assuming you and your opponent have both lots of time and resources, it's not great, but in normal situation when both of these are limited it's pretty strong.
You two haven't seen their unique item in action lol. Spam two 60 groups of that shit and no army could stand in your way,they are super op. No upgrades matter when you have +40 damage on them lol. Paladins die in a 4 hits,when it is a blob it means they die instantly. Also they have very fast attack and they could kill most units before they could react. Pikemen are no threat to them.
I know, I was disregarding the UU in the post, if we're thinking paladins it's not like Burgundians are hurt that much by no bloodlines.
UU is just so retarded that it's just not worth mentioning. It's not even something like Mangudai, which is powerful but you need to invest a ton to make it work and then without like 20 it's still not doing that much. Nah, with no upgrades coustillier is a knight -2 attack(of course if we ignore the elephant in the room of their charge) and 115HP(so -5 Frank/Bloodlines knight) for 5 less food and 20 less gold. So the bastards are going to work well even in small numbers and if that wasn't enough, as I've said before - Burgundians can just go knights/cav/paladins and expect good performance on their own(the economy is... reasonably good, probably there are stronger civs but Burgundians are quite high on the stack), and then they get crazy UU on top of that. I don't even despise the charge mechanic(and strength) that much, I think it's interesting, but it's put on a too strong of a base unit and given to a civ that otherwise was going to be strong enough even without it.