Craftsman transmission fluid question

I have just used a regular tacky red high temp bearing grease in those and plenty of un sealed gear boxes with great results for years now.
$3 a tube.

They didn't offer to cover the cost and I didn't ask them to. I've had the mower since 2009 so I it is long out of warranty.

Mine must have been an earlier version because there were some differences compared to the disassembly video you linked. The transmission in the video looked like it had some updates that I've seen on newer models.