Having only flown AA with my cat (they raised it to $150 last year too!), where they still require you to check in the old fashioned way (the website and app won't even let you check in), it's the type of thing that I could see holding things up if you get the wrong agent or a bunch of pets travelling at the same time. So far, I haven't had any snafus, but have had a few "could have become a problem" situations that DL's version avoids:
-As mentioned, I have to ding them on the OLCI bit, if only because that is an area they could save everyone involved time on IMHO (just flag the BP generated to make you swing by the counter to get the tag and do a quick visual on the carrier and animal meeting regs, etc., but still have the normal pax stuff out of the way would seem to be a smart middle ground)
-Agent is allergic and needs to summon another agent to handle my transaction (no biggie, only cost 5 minutes or so, BUT....)
-Agent decides to recite the policy manual like a Baptist quoting the Bible and wants to visually watch the cat do a 360 in his carrier (I know this is part of the regs, but eye roll)
-Ditto where they get out the tape measure and watch the scale like it's a wrestling weigh-in
-Agent has never done in-cabin pet paperwork before, so it becomes a dance among different staff answerings questions, entering this, getting that form, etc. (utterly forgiveable on the personal level obviously)
-By waiting until time of flight at the departure airport, the risk is there that you could get crowded out by the 5 pets per cabin limit -- AA does let me check a box for in-cabin pet when booking, but again I could see that getting weird if Murphy's Law hits!
-If you're not flying in J or F, or not elite, having to go through the general check-in lines means the usual hassle there compounding your own transaction, so a small nod in favor of upgrading more than you otherwise would to take advantage of priority check-in (at Xmas last year, I was behind a family of 2 adults and 4 kids off to India, with what must have been a dozen bags!
On the plus side for how they do things, I have had a couple flights where the agent looked past my carry-on that the rules say needs to be checked regardless when traveling with a pet, and said just bring it with me (I can't remember if DL requires you to check your carry-on bag). Even though I am a big fan of carry-on only to avoid the hassle of checking bags, when I have him with me, I have found the beauty of traversing an airport without a roller in tow!
As it happens, my flights with my cat have been where the AA fligths just work better schedule-wise, but ceteris paribus, I would probably lean DL for that more streamlined process. As someone who generally favors DL for their operational competence in such matters, I submit that my experience very much presents multiple potential points of failure!