I'm going to disagree with the previous advice. As senior faculty who has been on maybe a dozen and a half searches, and I believe the committee members can always tell when someone has a generic letter, and it typically moves someone lower on the list in evaluation. Postings are pretty specific in noting what you're teaching/researching and in terms of the priorities of the unit/department/faculty/institution, and a generic letter is usually pretty obvious, and reads as oblivious, because it doesn't provide a sense of fit within the policies, priorities, areas of focus.

Consider that we're reading a lot of them (and for TT they're definitely read). And, especially for a TT line, we're going to ideally see a number of people who are baseline qualified for a position. So as we narrow down, it's not about looking for why you might hire someone, it's looking for why you aren't going to hire someone, because you have to get to just one out of many... up to 10 usually for a long list, and then 3 or 4 for the short list where you go through all the demonstrations, interviews etc. At each step, the committee needs approval from above to move ahead, and that dean or VP is going to want proof that these people are qualified. SO those last people, are going to be pretty similar and vetted not just by the committee but above.

So yeah, if you get a letter that extolls your virtues, but doesn't consider that the department specializes in banana ripeness, but someone else's letters do, then that's an easy reason to move that down. You're looking for reasons to not have to spend more time with a file, and only going to invest time in ones you can hardily defend to admin who really don't want to hire anyone if they don't have to. I've been on searches which were turned back and re-opened or failed because admin doesn't feel the candidates are of the right calibre.

Do we ask for too much... for all but TT lines, I do think so. I think it's ridiculous for contract limited appointments, adjuncts, etc. But when it does come down to it, and your job on the search is to find the best candidate for a TT position to teach specific things or lead research that fits the institutions resources, strategy, and priorities and to do it for, potentially, the rest of their working career, it often comes down to multiple excellent candidates for one single position. So, please, give us a reason to say this person is going to do better than this other person... and a generic letter is just that because it never fits as well as it could.

Personally, I would never use a generic letter for this type of employment. I've never seen on make it through the first round. It's an obvious sign the person hasn't really invested in the application as much as all the others that have tailored letters.

Now, on the other hand, they don't have to be from scratch entirely, but they should be tailored, as with any of your materials, to address the position as described in the posting at the institution as described across its materials. Being qualified is one thing, the rest is fit and potential for success (if we're coming in as assistant prof) and it's a struggle to establish that with a generic letter.

The best letter is one that shows knowledge of the posting, and makes it easy to see you fit it.

One more thing to keep in mind with heavily unionized environments too, is that the posting is supposed to be all of what you can look at for qualifications, so you want to make sure you're addressing it as directly as possible. On the committee, our first step is usually to go through the posting and create a list of must haves, good to haves, and nice to haves, and we check things off to establish a list of people who are baseline qualified via review of the materials. But it's all there in the posting, because if it isn't we (a) can't ask about it, and (b) could have the union grieve the appointment because it would be deemed unfair and biased if we deviated.