Many don't. Can't be sure about all of them.

The three airlines I've been hired by didn't and the others I applied to didn't. Just ATP, yes or no and aircraft flown. No dates.

The actual ATP doesn't even list dates of type ratings. Just a date the current one you have was issued. I got my original in 1997 but my current one has a 2008 date (I think that might have been when we had to do the english proficient thing, but I can't remember for sure).

Bottom line is no one is going to care. Matching ATP dates with type rating dates by digging through logbooks, other airline training records, etc. is minutia. They can see from your employment history what your story is. It ain't rocket science.

Wayyy too much is being read into getting a 737 type rating, IMO. I know plenty of guys who got hired at Fedex, AA, UAL, Delta, UPS, etc. with 737 types. You get out of the military, you fill the ATP square, get a 737 type if you have the cash and shotgun the resumes to whoever is hiring. Airlines all know this and they'll know you're lying if you tell them you haven't applied to other carriers. They'll probably ding you for poor judgment if you didn't get apps in with everyone who is hiring. It shows a commitment to flying and the career.

The folks you're sitting across the interview table from are always your first choice - no doubt - those other apps are just in case it doesn't work out with them so you can get some more experience and re-apply at the earliest opportunity.