Some recommendations and additional info.
Segovia: The draw is suckling pig and several other local dishes. There are a whole group of restaurants all offering those dishes, with some variation. Someone mentioned Meson de Candido, there is also Jose Maria, both are well known, and you really could not go wrong. Last time we ate at Asador David Guijarro. Smaller place, dishes seemed a bit more right sized with a nice set menu with a few choices available, it was excellent, also had non-meat dishes for my wife. Tapas are really not a thing here, except a small complimentary snack in a few places.
Bilbao: We did have a couple sit down meals, the most notable at Restaurante Marisqueria Mazarred. I wanted the Spider Crab, this place was old school seafood, very well done, without breaking the bank.
Other than that, we focused mainly on pinxtos. El Globo was a nice stop in the afternoon in the downtown area. Then there are several places in the old town in Plaza Nueva. Gure Toki and Sorginzulo, next to each other were great, also Urdina and Cafe Bar Bilbao. There is a food hall type operation on the upper level of the market, it was OK, but had lots of variety.
You mentioned not seeing places open earlier in the week, first, you may be seeing current off-season hours, but yes, each place takes off one or two days early in the week. Also a bit of surprise for Spain, but in Bilbao, many of the bars and restaurants were closing up by 10 PM. So verify open days and hours when planning.
Hope to add some thoughts on San Sebastian later....