Download the parts diagram for you old transmission.
You will see 2 "shift keys"
these lock one ( hopefully ) gear on the layshaft to the layshaft and allow all of the other to spin freely.
Shifting on the go generally breaks the heads off the shift keys ) if you are lucky ) and if you are unlucky it locks 2 adjacent gears to the lay shaft.
Naturally the layshaft can not turn at two different speeds at the same time and one of the gears can break teeth or completely break apart.
If your English is reasonable download the Tecumseh/Peerless transmission repair manual.
It is everywhere on the web.
I got mine from here
https://www.mymowerparts.com/pdf/.
They give reasonable service when ordering spares from them as well.
Because Tecumseh went bankrupt a lot of the larger internet retailers will simply list all parts as NLA but most mower shops can get them and I get a fair bit from the UK, Just a pleasant Sunday afternoon drive from your place ( more fun on a motorcycle ) .
These are constant mesh gearboxes so every pair of gears is in contact with each other & always spinning . but only 1 gear should be locked into the output shaft at any time, or it goes bang & then dosen't.
The MST miht work for you but the big problem is indexing the gear positions so that the shift keys are as already mentioned only ever connecting one gear to the layshaft at any one time.
Different linkages on different mowers move the shifting lever different lengths .
Some boxes has a stiff indexing mechanism so there is little to no shift gate on the mower while others have minimal shift indexing in the box and rely on the gate in the mower to keep the lever in the right position.
