If this MTD has a hydrostatic drive transaxle, there will be a "free wheel trans" lever somewhere which will have to be moved from the normal drive position to the "free wheel" position in order for you to be able to roll the wheels on the ground.
If you find this lever or knob, and moving it allows you to roll the tractor on the ground, the hydrostat is at least functioning in the "brake mode", but as you suspected, the only way to evaluate the transmission is to use it.
It's going to be up to you how you decide to spend your money, and what you consider to be a reasonable sum might be way too much for other folks.
My personal opinion is that if you are going to spend more than $200 to get the engine going, you spending too much.
If you could get the engine running well for $200, and the tractor was in operating condition after that, you would not have hurt yourself much.
The problem lies in just what you posted about........unseen/unknown other issues that you may have to address after they reveal themselves in normal use.

I bought a Murray lawn tractor that had problems. Has a good engine, nice tires, great seat, nice wheels, good hood, great battery, and doesn't move without groaning--a lot! $50 is what i paid. The battery reposes in another machine, the seat just went down the road on an old Craftsman tractor, and the rest will be used as needed. The engine will repower another machine i have here. Tires/wheels will be sold soon. Didn't lose much, if any, on it. Got lots of good parts. And, the scrap yard will buy what is left!

MTD made a relatively few lawn tractors with hydrostatic transmissions. Most had continuously variable "automatic" transmissions. You need to look underneath the machine and under the hood.

A hydrostatic will have a single belt going back to the transmission, using a belt tensioner arrangement as a clutch. There will also usually be a fan under the pulley on the transaxle for cooling.

The CVT system or "transmatic" uses two belts to connect engine to rear axle. The front belt connects the engine to a stack pulley through an idler-clutch arrangement. The stack pulley is then connected with another belt to the rear axle. The rear axle is simply a forward-neutral-reverse assembly, and speeds are controlled by the two stack pulleys sliding apart to change the belt ratios.

A hydro will have either a foot or hand control which adjusts both speed as well as forward and reverse. The transmatic will have either a fender, dashboard, or tunnel mounted speed control; and a separate forward-neutral-reverse lever.

If your machine is a hydro, the second post is probably right, and things are going to cost quite a bit to get the hydro right. The transmatics are actually pretty robust except for a propensity to eat belts. However the rear axles (which are Peerless) do have a habit of locking up due to play in the shift lever and keys. When that happens, things ccan also get expensive in a hurry.

Thanks for all the good information. I have mowed with a MTD transmatic for 15 years( very tough mower) and have been very pleased. I saw this automatic version of the MTD i already had and thought it would be nice to have an automatic