Oh my, yes the original oil in the gear box has broken down to the point of becoming a solid. Rosie utilizes a variation of what is known as the Beam transmission, which was used by several manufacturers within their own private label wringer washers, as well as automatic variations.
A few years ago I restored a Speed Queen automatic washer, which uses Speed Queen's variation of the Beam automatic transmission, which is quite similar to Rosie's as a wringer/conventional. During the restoration process I researched through several different brands of the same Beam design to figure out what oil was originally used. I eventually stumbled across one manufacturer that called for a 90w straight gear oil to be used. In 1950's talk, 90w is what is considered today's 90W GL1 gear oil. The copy of your service manual is calling for a 30w motor oil, which on the ISO viscosity scale is very similar to non detergent 90w GL1 gear oil.
I'd recommend that you guys clean out the transmission with mineral spirits and go grab a bottle of GL1 90w gear oil at your local NAPA or old timey auto parts or farm supply stores. Don't go to a newer Auto Zone or similar auto parts store - they won't have it, and do not become tempted by a detergent blend of 80W-90, or some other variant with two numbers in the specified weight.
Fill it to just about the top of the main drive gear, driven by the pinion gear/shaft. On a wringer, I'd guess that is around 18-24 oz. Re-use the gasket, if possible. If the gasket is shot, scrape the surface clean and either cut a new gasket from 1/16" material (from the same auto parts store you found the gear oil from) or seal the case up with a very thin line of RTV.
I've included a link below to my '67 Speed Queen restoration, where I document the trials and tribulations of figuring out what oil to use.
https://automaticwasher.org/threads/a-royal-restoration-summer-of-love-speed-queens.92884/
Ben