Making sense financially and making actual logical sense from a self-preservation point of view are 2 different arguments. You argue the former, I argue the latter, and with good reason. What's cheaper, a cheap motorcycle or an ER visit with a couple of days in the hospital. I know the answer to that without even having to think about it. My last ER bill was over $10k, and even with insurance I had to pay $2500 of it - and that was 3 hours with no major injuries. Bicycle frames are designed to handle the power of a human (less than 1/2 a HP for the average person), not 20 times the power of a human. 2-4hp from a gas engine is within reasonable limits for most bikes, but even those can cause the tubes and welds to crack and fail, not only from flexing due to weight, but the vibrations.As to @ImpulseRocket response, I did already mention in the original post that it's not really legal to call it a motorized bicycle but you could try to get it registered, but then at that point an enduro makes a lot more sense to get or any street legal bike. It'll cost less money if you already have a bicycle and just stick on some mirrors, headlights, and a license plate with whatever engine you pick to get to far places or generally drive safer where there's no sidewalks and high speed limits (even where I live which is 10-30 miles from where the majority of the businesses are at)
It makes a lot more sense to get a engine thats made for dirtbikes because you can limit yourself to how you handle your bike, just like with cars you wouldn't step on the throttle even if you have a hellcat but I'd still only do this on sturdy frames and opt for wheels with bigger hubs and spokes and a fat tire preferably, which could be already done with a pretty decent mountain bike which many are designed to handle high impact. You can get hydraulic brakes for pretty cheap which is sufficient for most roads and you can also upgrade your suspension, there's people who take mountain bikes and go hurling down a mountain doing jumps so for high speed I wouldn't say it would be a concern, must make sure your hub is well, built and opt for bigger tires.
I haven't used a 2 stroke dirtbike but I've used 4 stroke on the road before and as to the 4 strokes there's no such thing, most to not all dirtbikes come with a shifter so you can build up your rpm before you shift, a bicycle will certainly be more lighter than a dirtbike so you should get a nice boost to acceleration, even my mini dirtbike weighs more than my phatmoto but beats my go kart (stage 2) in acceleration, just not in top speed because the predator just does not care what frame it's on. I'd imagine a 2 stroke engine meant for a dirtbike would outdo any Chinadoll but I'd stick to 4 stroke simply for the torque, stock should get you 40s to 50s and 80 or 90s is excessive and illegal to even drive anything at that speed. Just I said "looking for raw performance" but we shouldn't leave out mid to low ranges and I'd rather trade that top speed for torque.
After thinking about it, I'd say the most practical setup if you have some mechanical prowess is a well built mountain bike with the generic 125cc 4 stroke you can easily find for 150 (excluding Chinese websites.) Shouldn't be too expensive to upgrade its parts and stock performance is great, they're meant for mudding so they should be able to tank anything that comes from the road and they should be pretty reliable granted you take care of the fuel. Depending on the state you should be able to spend the extra money to get it registered or if you want to be risky just drive it as is, if you got a regular bicycle or a cheap bicycle I'd say it'd make sense to go with the Chinadoll as you dont want to go at high speeds In the first place or just destroy your frame or really anywhere you'd rather stay at low speeds just go with a chinadoll. Although the chinadoll has potential to even outrun these other engines stock that's not without paying almost double the price of the kit excluding labor. Also with the China doll you'll either be sacrificing high-end for low end performance or vi versa. Really depends on the persons situation, china dols are perfect for the typical consumer who wants something they can slap on their bike to go legally fast but i dont really see a point in spending the money and time to tinker an engine to perfection to go as fast as something else at stock for cheaper, the entire design of the china doll is for long term reliability at low speeds, same situation with the predator and its original honda gx, meant for workloads not rideability. You can make 2 very simple changes to the predator thatll let it gain 25% more performance which should tell you its purpose. If you just want to have fun just take a dirt bike out for a ride, if you want to be able to go to different cities then id say equip a mountain bike with a decent engine and get it registered, if you just want to ride around locally then just get a china doll.
Not all dirt bike engines have transmissions. the 50-65cc class in particular is usually just a wet clutch drive and a single speed. The KM 50sx and 65sx engines I named are as such.
I've ridden 2 and 4 stroke enduro and dual sport bikes, and still do. Very familiar with both. 4 strokes are much much much better suited for low speed and street riding specifically because the majority of 2 strokes are tuned for peak power output, which means they lack a lot of power under the curve, and tend to make everything up top, thus... ____/ style power curve.
I'm not trying to tell you not to do something, so don't take it as such. Just offering the counter points and wisdom to temper ambition.