sorry if this is a bit long - I'd usually edit it down but am in a hurry.
So I actually am having trouble following your example.
1 - Do you mean that the landed contract is actually really big and important (basically it's now paying for all of the development) or just another signup?
2 - you called them hyperbolic but picked two examples that can keep clients from taking the software seriously: both examples seem to be something that might be much more important than whatever the team was working on.
What I mean is that if the client is trialing the software, and has a certain process in place but trying yours, then if their search is too slow they might go back to whatever process they had before; you can lose the client. Likewise blurry profile pictures might cause them to decide that it is not polished software they can use.
You did not use examples like "unclear icon on search label!!!" or "client logo NOT on their dashboard" or something stupid.
I also thought that anyone could talk to the stakeholder who added to your board and figure out why it is so important (if it is) or immediately move it if it's not (Assuming it's a big, important client.) Or edit it as appropriate.
Obviously this might mean totally rewriting the feature: perhaps slow search can just be rewritten as showing a recent list of the last ten (whatever) and allow them to be pinned, as the client doesn't really care about search but just wanted to bring up (whatever). I say this because for really old searches from archives people wouldn't care about waiting a few seconds, whereas for workflow type things done lots of times all the time it really matters.
Obviously a PM is supposed to stand between such a client/internal sales director and the developers but I am having trouble understanding why the PM can't move and translate from the board - assuming these issues and the client is big/important enough.
Emails often get lost or not handled with appropriate urgency for the case at hand (and can even be marked wontfix or feature request simply due to lack of communication) so if you wrote a bit more about your example I might understand better.