Surely I'm not being told I need to abandon my whole BUR and submit each alias as a separate whole-ass thread just because there are other tags already aliased to them, right? I can't imagine why that would be a useful feature.

The error messages exist to avoid potential site breaking bugs and similar issues. Could you paste the script here?

It was a long one (so long that I'd have to break it up anyway), but the relevant lines were:

alias purple_inner_ear -> purple_inner_ears # has blocking transitive relationships, cannot be applied through BUR
alias grey_inner_ear -> grey_inner_ears # has blocking transitive relationships, cannot be applied through BUR
alias brown_inner_ear -> brown_inner_ears # has blocking transitive relationships, cannot be applied through BUR
alias tan_inner_ear -> tan_inner_ears # has blocking transitive relationships, cannot be applied through BUR

Hmmm, where to start...
One, The correct tag would be [color]_inner_ear, not [color]_inner_ears. That's why a derivative tag is called inner_ear_fluff and not inner_ears_fluff

To address the question in your title, a BTR occurs when the antecedent tag in the alias you're suggesting has an active/pending alias/implication request. If you metasearch the antecedent tag, and you can spot the word Active or Pending, you're gonna get a BTR.

The reason why BURs have BTRs is because BURs have built-in safeguards because of what they can do, or more accurately what the system can't handle doing. BURs aren't that closely intertwined with the alias (and implication) request system (and sometimes BURs feel like an afterthought) and that means they don't have access to certain features.

Oh and BURs were initially created as an admin tool, and only recently released to the public, so that may explain some of the strange design decisions

I could swear I've seen BURs where some of the tags they're proposing to alias already have aliases to them.

Heck, I just went looking for one and found a BUR that's the exact opposite of mine; shouldn't that have blocked ALL my lines because they're marked "pending"?

errorist said:
I could swear I've seen BURs where some of the tags they're proposing to alias already have aliases to them.

Errors don't show if the BUR has been approved/rejected, and if there's no warning while the BUR is pending it's because the aliases and implications on the antecedent tag were added after the BUR was made. The errors would still be there, but they don't show because the BUR hasn't been manually updated yet (the creator of the BUR can just go to edit, do no changes, hit submit, and then the errors would display themselves).

Heck, I just went looking for one and found a BUR that's the exact opposite of mine; shouldn't that have blocked ALL my lines because they're marked "pending"?

Only alias & implication requests (the individual requests, which is not part of a BUR) block other AIBURs, BURs do not block anything while pending.

The best tool to find if there's a BTR is metasearch. If there's nothing showing up as pending or active for the tag you want to alias, you will never get a BTR. If there is anything, it will result in BTR.

I'd recommend checking your BURs with bitWolfy's debugger: https://bitwolfy.com/projects/bur_debugger/
It's a pretty nifty tool that will tell you immediately what issues your script has, if any.