Were it me, I would have started with a pre-2000's Craftsman mower as a base. They have a 6 speed transaxle with a differential (which solves the steering problem mentioned) and a built-in brake, and examples with broken or missing gas engines can be had used for $100 or less quite often. They have that boxy sheet metal look of old tractors too. It would also be possible to adjust the pulley ratios to slow it down or just block off the higher speeds until the kids get a bit older.

Granted, I understand that the purpose of a project like this isn't just in the end result. Depends what crafts you want to practice and what's just necessary work around them. There's still quite a bit of fun project left in converting an existing mower to electric and refinishing it to look more like a classic tractor.

But those are internal combustion engines, so each time your kid want to have fun, it annoys everyone.

I'd rather have my kid ingrained with the idea that electricity is the future even if it's an amazing achievement to be able to tame explosions to move around

You can buy those transaxels (different number of speeds) surplus fairly cheap, but you have to look.

Though the goal was a kids toy, and those mowers are too large for that use.