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.