Well, thought I would add an update.
For the last 6 years, I've been flying Tpac a couple of times per year (until Covid) and occasionally, it was more convenient to have separate tickets.
As a 30+ year Medallion, I always fly SkyTeam, unless it's simply not possible.
In the last 6 years, I have not had trouble getting the check in agent to tags bags to final destination.
Although, the last time, just last month, December 2022, leaving AUS to SFO on DL ticket and metal; then about a 12 hour connection time to a non stop Vietnam Airlines flight SF) to SGN, it took a few check-in folks in Austin, to get it done.
First response was that it needed to be on Delta. That was obviously wrong and I told him in nice way, that I have done in a number of times and never had a problem (true).
He got his supervisor and she was also skeptical, but was willing to listen.
At that point, the nearby check-in (this is all in the Sky Priority lane) agent, chimed in and showed both of the other agents how it was done. took a few minutes.

I was ever so grateful, as my wife was traveling alone with an embarrassing number of exactly 50 lb bags and did not want to contemplate her trying to get the SFO shuttle (which she did not know about at this point, and I wasn't going to tell her) alone, with those bags.

In sum, it can be done with Sky Team airlines and it's one of the reasons I am so loyal.
The other reason is how these Delta folks really tried to solve the problem without making it about egos.

Delta is great because of the people and it's why they have had my business all these years.