I had some time to experiment some here. It had been about 18 hours since last driven and right off, CarPlay failed to open up properly. In my case, it failed when locked or unlocked despite repeated attempts which included plugging my iPhone in and out; trying another cable; and "forgetting" this car and setting CarPlay up again. Ultimately I was able to get it to work properly by doing a hard restart of the iPhone, but this process won't cut it long term. That's what it took the previous times this happened, and I don't think even that worked the first try on one of those times.
I decided to do a complete reinstall of iOS 26 with a restore of a backup made right beforehand since at this point that's all that's left to try. CarPlay did launch correctly when I retried it, but since the iPhone was freshly restarted as part of the reinstall/restore, I'm not prepared to say this cured the problem. I'll report back in a few days, sooner if the problem recurs.