It turns out that the entitlement approval process is two-stage, even though I can't find anything documenting that. The Certificates, Identifiers, & Profiles support people weren't helpful. I filed a TSI, and they told me I needed to submit the entitlement request form a second time. I did that (I put something in sponsor field about wanting Distribution access), and after a few hours I got another approval mail (identical to the original one I received). After that, my distribution profiles had the entitlement.
ifdev
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Update: The entitlement migration people were not helpful. They just said to be sure to request approval for the entitlement, which I've already gotten. I've put it in a support ticket now.
Anybody make progress with this? I tried submitting the entitlement migration request, but that really doesn't seem like it should apply to my situation since I'm not use XCode Cloud. Development profiles get the entitlementment, but ad hoc and app store profiles do not get it. I have tried entirely new profiles, certificates, etc. I have double checked the App ID, and it has the capability enabled, and the capability says it is available for Development, Ad Hoc, and App Store.