The Crucible

The Crucible has some nice Family classes, exactly what you are describing in terms of age range! If you look at their classes on offer and use the “Age” filter and select “Family” you can see what they have on offer! I haven’t taken the family classes myself but I’ve been having a great time doing ceramics classes there. It’s fun going there because you sometimes interact with some serious artists and can talk with them about their work sometimes. 
 

https://www.thecrucible.org/course-search/

Audrey

I'm a fan of the Crucible's industrial arts programs for teens. My daughter loved their leather-working class, but they have welding, woodworking, glass, kinetics and electronics, ceramics, and more. Very Oakland. Accessible by BART or bus, if you don't want to chauffeur. https://www.thecrucible.org/

The Crucible (near W Oakland BART) offers youth classes. https://www.thecrucible.org/classes/youth-classes/

The Crucible (https://www.thecrucible.org/classes/youth-classes/) has great programs for teens. Their focus is industrial arts, including welding, glass blowing, woodworking, and ceramics. I've done adult classes there (jewelry making) and my daughter did leatherworking camp multiple times.

The Crucible runs great 1/2 day industrial arts camps during the summer. I think it's all week-long camps, so timing is flexible. The downside is cost, as they are on the expensive end (but you get a lot for the money). He could learn welding, leatherworking, or glass blowing, among other things. It's a block from the West Oakland BART station and on the 62 bus route so he could manage his own commute.

You could contact the Crucible at The Crucible.org. Their woodworking faculty are really good at making interesting furniture with wood and metal.

If he has any interest at all in any of the industrial arts, The Crucible in Oakland runs awesome classes for teens  https://www.thecrucible.org/product-category/class/   One of my kids attended a camp over the summer and their covid protocols were excellent and consistent, and he met a few like-minded kids.  He's even met a friend from that camp a few times to hang out - the crucible seems to attract some pretty cool people.  It's not just for kids - in addition to the camps, they run weekend classes year-round for older teenagers and adults.  

I don't know what you consider affordable, but the Crucible offers half-day arts classes open to kids 8 and up. My daughter has done multiple week-long summer camps there, mostly leatherworking, and it seems to be in line with the way you describe the carpentry class.

https://www.thecrucible.org/classes/youth-classes/

The Crucible in Oakland has all kinds of cool classes in things like glass-blowing and metal-working.  It’s a short walk from West Oakland BART, so she could probably get herself there and back.   

A few years ago I took some woodworking classes at The Crucible. I met some really interesting people, learned some new skills, and got a beautiful table and jewelry box out of it. The Crucible is expensive, but Laney has a well respected woodworking program (although I don't know how hard it is to get into). Good luck!