Not quite.
The need and requirement is for a (technically and certification)competent assessment of compatibility to be made as the STC holder, patently, can't do that for individual birds.
Clearly, there will be many installations for which this may be straightforward and the installing LAME can do it just fine.
However, and particularly as we see increasingly electronic birds, it just doesn't work across the board.
For instance, a mate runs a fleet of highly modified, fitted out with more kit than you could poke a stick at, aircraft ... which regularly drive the whitehanders and design engineering support guys to distraction with the innovative gremlins which pop their heads out regularly.
The LAME who allows him/herself to be browbeaten by a customer into doing a down and dirty assessment when such is inappropriate .. puts his/her neck into the noose whilst hanging the fundamentals out into a cold, cold breeze ...
Hard facts of life, I'm afraid.
Same thing applies the other way around - the designer of a one-off mod intended for a particular tail number has to ensure that it is compatible with both the OEM bog standard aircraft configuration plus all the other existing mods, including STCs, which might be present in the aircraft. Sometimes there are hidden things there which can bite everyone and even the experts goof off on occasion following which there is a serving of embarrassment all around ..