have you applied to jobs advertised in job listings by game studios (large and small)?

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.

I know a handful of non game programmers who just applied and their combination of enthusiasm for games and general engineering chops got them jobs at Blizzard.

I know the remote aspect is important to you, but having done this exact switch (general backend services dev -> backend dev for a big game company) this requirement is almost certainly going to hold you back. The field is bleeding jobs and there are plenty of people who will go ass-in-seat 5 days a week. Outside of that, there are a surprising amount of roles that generalist SWEs fit into that don't require any experience. It's been a very comfortable foot-in-the-door for myself, at least.

The game industry inherently unsafe. Studios go in and out of existence constantly.

Please continue to make your minigames in your free time and don't step into the games industry unless you know what you're getting into.

I have friends in the games industry who also went into it because when they wanted a job to make and play games, and now they are extremely unhappy and depressed.

Low pay, crunch times, constant strikes, layoffs, micromanagement is all there in this industry.

Unless you own or found a games studio, being an engineer in games industry isn't what you think it is.