After all, millions and millions and millions of rolls of Kodachrome were quite happily processed in roller transport machines!
I don't think so; I think they were more than likely cine machines.
In the lingo of the trade, a roller-transport machine uses a large number of semi-opposed rollers that can essentially self-feed a piece of film.
A cine machine, on the other hand, is a leader-fed continuous processor. The idle machine is fully strung with leader. When you start it up you attach the front end of the film to the leader, and the leader guides it through. All of your film is spliced together, and at the tail end of the processing run you attach leader to the film; this restrings the machine with leader again.
One big advantage of the cine machine is that the emulsion almost never touches a roller, only in a couple of places. The way this works is that each rack, the frame that sits down in each processor tank, has a set of rollers on both the top and bottom. The film is wound, in a single strand, around all of these in sequence, with the emulsion facing out. So the emulsion virtually never has to touch a roller. And where the base contacts a roller, it usually rides only on the edges, or on a dimpled rubber sleeve. (Cine machines can run with very little tension so it's not a problem to have dished rollers.)
Another advantage of a cine machine is that it can use "proper" squeegees. The leader can feed the film right through. A roller transport machine, on the other hand, can essentially self-squeegee by squeezing the film between rollers at the exit of each tank.
Fwiw user laser, on this site, can probably confirm the Kodachrome machine type. He was, as I recall, the chapter editor for "Processing Methods" in the last edition of IS&T's Handbook of Photographic Science and Engineering, which means that few people would be more knowledgeable about general processor drives, etc.
Sorry to be so pedantic, but there really is a big difference between the two machine types. Roller transport machines can do really good work if they are scrupulously cleaned and maintained, ind if the specific film can take the pressure, etc. But the ultimate, in my view, is a properly set up cine machine.