White Mountain would be great on a mountain bike. There's a road all the way to the top. It's a very rough road that only a specially outfitted jeep could negotiate, with an experienced driver. There's a stone building at the summit with electronic equipment, so it needs to be accessed from time to time.
First, you have to drive along a bumpy, rocky, 16-mile long dirt road to get to the locked gate, where you park and then camp for the night. The "road" to the summit is very, very rocky with some big rocks. It's ride-able, but you might take spill or two. I think a lot of people ride to the top. I'd like to try it sometime.
As for Shasta, I've never used a guide there, but if I did, I would use the local outfit, Shasta Mountain Guides. The Avalanche Gulch route is by far the most popular, but there are other, even more technical routes. Check out all the options:
https://shastaguides.com/