Bottom line, the rules we use for the sanctioned clubs (ARMA, SOMA, USLMRA, WOO, etc.) have the safety of all concerned at heart. Building to the rules of one of those clubs affords a margin of safety and is a great way to teach young people about fair play. Even if you don't race on a national level the rules should still apply to your builds. Get the other Dads on board with one of the national club rules too. This way you can know that your son is racing against safe machines as well. The USLMRA offers two classes that use identical machines. One class, known as JP is for younger drivers, the other IMOW (International Mower Of Weeds) is for adults. The idea here is that one machine can be built for two people to race. A pretty cheap way for Dad and Son to share a Saturday. The other clubs offer similar classes, they just go by different names.
No matter what direction or set of rules you follow; There are three 'must do' items.
Steering. Stock steering simply WILL NOT be safe.
Brakes. Again stock brakes WILL NOT be safe.
Kill switch. With a tethered key.
There are many 'old school' tricks that are simple and effective. You do not need expensive hydraulic brakes nor do you have to have a complete custom front axle and steering assembly. This sport has developed many parts and pieces that are billet this and solid that, and that is fantastic. But a welder and some thought can build a safe, fun to drive machine.
Oh and simply changing pulleys, nah, not a good idea. The rules in USLMRA for both JP and IMOW have an 8 to 1 final drive ratio. With stock motors and stock RPM (another requirement in that set of rules) you will see speeds around 25 MPH. That is extremely hard on stock steering and brakes and they WILL fail.
Are there any clubs in your area? If so visit them, look over the machine to get ideas and search the database of knowledge on this forum. There are many folks here that will help as long as they know your build is safe and follows some sanctioned rules.