I don't think there's anything wrong with accruing Delta miles. That's not incompatible with the other option, getting an Amex or Chase card. You can use those card(s) for your spending, and still accrue miles on your flights. If you really want Delta miles, you can get a Delta credit card and earn miles on all of your spending. If you want flexibility, you can get the Chase or Amex cards you mention, which transfer to a variety of partners.

By the way, you may not be limited to Skyteam airlines because KE is in that alliance. It looks after a quick search like any DL/KE flights from RDU to PUS is going to require two connections. If you're making two stops anyway, then there isn't necessarily much of an advantage to flying KE, and you could fly another airline. For example you could fly on AA/JL, going RDU-ORD-NRT-PUS. Flying United, you can go RDU-SFO-ICN-PUS. And you may prefer UA because it has nonstops to SFO (of course, if you want to accrue more miles, you may prefer more stops, not fewer!).

If you flew to PUS once and SFO twice you'd have enough miles for a free r/t in the U.S. So it's definitely worth racking up some miles. If you prefer DL, there's no reason not to accrue DL miles. They do get a bad rap in some contexts but I've had a pretty good experience with them. The only reason to hesitate would be because it's rumored that their award program might get revamped (in a bad way) this year. But that's just rumor. If you do like them, then I'd go with it.

The only thing to make sure is that, whatever you pick, you make sure that all of your flights are with that alliance going forward. When you don't fly all that much, it's annoying to have 10-15k miles in two or three different accounts.