the Burali-Forti paradox saying that the predicate "well-ordered" cannot be self-applicable ... I just have trouble believing that there's actually any such thing as an uncountable ordinal out there, because it implies an absolute well-ordering of all the countable well-orderings; it seems to have a superlogical character to it.

I don't think that it's fair to characterise the B-F paradox this way. The argument of B-F is that, given any collection S of well-orderings closed under taking sub-well-orderings, S cannot be among the well-orderings represented in S itself. There is nothing paradoxical here. (I'm not sure whether this matches the content of Cesare Burali-Forti's 1897 paper, which I haven't read and of which I've heard conflicting accounts, but the secondary sources all seem to agree that he did not believe that he had found a paradox. ETA: After following the helpful link from komponisto, I see that sadly this is not how B-F himself viewed the matter.)

Now, if you add the assumption of an absolute collection of all well-orderings, then you get a paradox. But an absolute collection of (say) all finite well-orderings leads to no paradox; we just know that this collection is not finite. And an absolute collection of all countable well-orderings leads to no paradox either; we just know that this collection is not countable. And so on.

Of course, none of this shows that such collections actually exist. If you said that you don't really believe in uncountable ordinals (perhaps on the grounds that they're not needed for applications of mathematics to the real world), I would not have commented (except maybe to agree); but calling them incredible (as you seem to do, counting them as evidence against set theory, indeed among the strongest that you know) goes far beyond what I would consider justified.