My sister & I had limited screen time as kids (Friday night, Saturday morning, maybe an hour after school - maybe). Otherwise, my rules (and circumstances) were very similar to yours. Home computers were not a thing then.
Now, for my kids, we do Friday night and Saturday morning TV. They know they get this & anything else is "extra". They can lose Friday or Saturday if they behave badly. They watch Netflix streaming and PBS kids via Apple TV. Sometimes we buy shows. We don't do commercial TV. Saves a lot of hassle! My husband and I take shows off the "allowed" list if we see bad behavior after the kids watch them.
Saturday night is video game night. Same rules for behavior, and the games are still usually purchased by me and my husband, so no unapproved games.
The boys do get to watch shows or play games at other times during the week, but that is dependent on what my husband and I need to get done. That's always "extra" and we, the parents, say if it happens. There is no reward system for screen time during the week, since it's usually not gonna happen.
I guess it seems kind of strict as I write it down but we see lots of behavior problems around screen time. My oldest is only 10, so we'll see as he and his brothers get older.
ETA: When my boys get TV time, they each get to pick a show, one to 3 shows each, depending on the time. The shows without commercials are usually about 20 minutes long. My husband and I watch "our" shows after the boys go to bed.