The shift pivot bolt is probably the culprit. Loosen the pivot bolt jam nut at the inside of the pivot 1 turn..........try turning the bolt CCW one turn. Then CW one turn. If the shifter moves with the bolt rotation, you have found the problem. The pivot bolt is threaded into the frame and has a jam nut locking the bolt in place. Remove the bolt and apply liberal amounts of grease to the shifter pivot bore n reinstall.........Good luck !
You will be able to verify this when you loosen the jam nut and start removing the pivot bolt. It will pull the shift lever downward hard enough to engage it in first gear and when you reverse rotation of the bolt it will move the lever upward...
The first thing to check is the pivot bolt the shift lever is mounted with. It will rust and cause the shifter to stick. I had to remove and liberally grease it to keep it from sticking. * The pivot bolt has a nut on the inside and the bracket it goes through is threaded so loosen the nut and hold it while removing the bolt.
There are way too many factors (tire condition, miles, riding style,etc.)to tell you what you need to do........ Each is an individual case that only you can decide what is best for you .
I don't think Hylomar is made for your intended purpose. New 0-rings & silicone sealant I think is what you need.Make SURE there are no cracks or warped surfaces. Can you post pics?