Leave the "patina" of it's former life and just freshen the parts that need it.

another vote for leave it as it is....
Perhaps repaint the iron but leave the wood with all the marks and stains.

I agree with the above. It's a little slice of history: let it tell its tale.

Perhaps a light cleaning with 000 steel wool. That should also smooth the surface of the wood. Then you can use some rub on polyurethane to seal it without giving it a plastic look with thick brush on polyurethane or varnish.

If it's similar to all the old wood school desks I have seen and to the 3 I own, there are many many coats of varnish, as most schools covered the desks with new varnish every year.....and you would have to remove many coats to "smooth the surface of the wood" using OOO steel wool.
Enjoy the history and think of all the kids carving their initials with a protractor and getting ink stains as they learned to write using the Palmer method.