Pros cons? Obsolete? Couple of rifles I'm thinking of. The 22-250 is a savage axis and the 222 I'm waiting to hear more about. Reloading is probably the only real option for either.
The .222 brass isn't as common, but far from obsolete.
I believe most of them don't have an overly fast rate of twist similar to the 22-250.
Depends on what you want. The 22-250 is in a whole other class. It's around 700 FPS faster with a 50 gr bullet. But again limited a bit by having to stick with lighter bullets.
You mentioned that the LR primer use was a plus, but to me these lately are just about twice the price, and harder to come by.
.222 would be a dandy lightweight walking varminter with milder report, long barrel life, and cheap to feed. But if you want to zap stuff at longer range, the 22-250 has the edge.
Everyone needs a mild walking varminter small game gun, so I like the .222. And maybe a longer range varminter too... but I thought I wanted a 22-250, until I realized the .243 can do everything it can, but better, and also an easier stretch into big game.
About 300 FPS faster with a 55 gr. and going up to 65, 75 gr varmint bullets can really leave it behind at long range.
I'd forget the Axis.