You don't 'need' the remote to operate the set or any of its functions (Zenith was good about that).
Don't worry about the remote until you know the CRT is good on all guns. A new remote is $10-50 a new CRT is $150-300.
One of these is probably right.
https://www.ebay.com/itm/ZENITH-SPAC...gAAOSwhplaUs4J
I can't recall seeing a Roundy with anything less than a 400. 600 sets have a motor on the back of the hue pot shaft, 400s don't. You don't need the remote to use the set. Be sure the set has its remote chassis (separate from the main chassis, tuner cluster, and convergence)...If it does not then it is either a Non-remote motorized tuning model or is missing its remote chassis.
Also Zenith remotes are ultrasonic there are no electronics in the hand unit, but rather tuning forks (really more like rods) that the buttons strike. Zenith liked to play with the order of the rods VS the buttons each model year so if the buttons are all working but labeled wrong then try opening it and swapping the rods around.
It would be a pre-July 1964 set based on the lack of UHF. The government mandated all sets have UHF in mid '64 before then it was optional equipment.