Our experience two weeks ago with Delta. Husband is Diamond medallion. No issues at the ticket counter to check in using the priority line.
The customs lines are another thing.
Delta will tell you they have their own line - line #4. Lines #3 and #4 feed through the double doors single file which makes the Delta line pretty useless. We jumped out of the Delta line and into the line for non-passport holders - just drivers' licenses - even though we had our passports. We beat the people who were just in front of us in the original Delta line.
The global entry line was really short - but specifically states you need to have your physical card (I had my global entry card, but husband didn't bring his card with him)

Use your passport or drivers' license or global entry card and pick the shortest line.

FWIW - As for the red caps - our kids went through a few days before us and tried to use one. They were told the service was only for those checking bags.