Not to disappoint you, but a career in the games industry will be vastly more difficult than a typical SWE role. You'll be expected to crunch for a game's launch with no guarantee that you won't be laid off after. There's a steep pay decrease for roles with the same YoE.

Skills wise, I would recommend spending a few years learning C++ at the least, try developing an engine from scratch, build an ECS, build some basic multiplayer implementations. I wouldn't expect much of your web experience to transfer, and you would most likely be joining the industry as junior.

Stability wise, if you have any backend experience, applying for infrastructure SWE roles that handle online services for video games is an alright bet, but that job market is quite small.