Having been through all this, I would most of all prioritize getting a permanent residence permit. This brings stability, and then you can decide whether to work in games (more fun) or elsewhere in tech (higher pay). Having been in both industries for a long time, I can tell you you’ll always wonder if the grass is greener on the other side. But at least you’ll have options. If a tech company gets you to permanent residence quicker, go for it.
Oldest game I remember playing (when it was new) was Lemmings on the Macintosh. Early 1990s. I can still hear the tunes.