The best advice I can offer is some my mom offered me in situations like you'res. "Doing nothing is often the best thing you can do".
You wrote;
outside many years holding up suprisingly well
I like the weathered gray look
It will never get better than that but with your help,it can get much worse. Anything you apply will discolor , peel and chip with exposure to the elements. Emagine what the chair looks like then.
White Oak stands up to the elements amazingly well while takng a patina impossiable to replicate by other means. Cypress , osage orange ,teak are a few others that do the same.
Think of it like this. A young woman with natural beauty applies a bunch of hollywood type makeup before perticipating in group outdoor sports.
Grab a tall glass of ice tea,lean back in your wonderful old chair and injoy the day.