It's 60 years old...!!If you want a new table, buy one.
I would rub it down with a very fine abrasive...like a water and pumice slurry...just to even things out. then I would rub with an oil with dryers in it....and repeat...and repeat.
Enjoy the evidence of your living with that table. It's a well loved friend, no need for it to look new, just well cared for.
Linda c

One option we considered is simply restoring the oil finish (after removing the water spots somehow) and topping the table with glass for protection.

How would you go about evening out the discolorations and water marks?
I would get a good brand of 'teak oil' from a reputable Scandinavian furniture retailer, some 0000 steel wool and systematically oil and rub steel wool all over the entire table (top and legs and apron), wipe off the oil with a soft cloth and repeat.

Skip the glass, and just oil the table top every few months. They develop a marvelous patina and are nearly indestructible.

We are also considering refinishing with a good quality waterbourne poly, which would mean we could use the table like normal. Until the poly starts peeling off in sheets. Teak is a very oily wood and finishes do not stick to it very well. That's why the classic teak finish is oil.

Bring it to me and I'll see what I can do with it.

Muhahahaha!

Alright, why does everyone think glass is so bad? I think every table should come from the factory with glass already fitted. As long as it's manageable to get clean underneath it if a spill goes bad and it's not so tight it can't breathe, why not? (Keep in mind I have a vinyl table cloth duct taped around MY table.)

Why not call someone at David Rago, Rago Arts and see what they would think about refinishing it.

Its still a piece of vintage, manufactured, furniture not an 18th c. piece, so some people feel the original finish for such things is important only for very early or prototype pieces. But why not ask the experts? Art Deco furniture is refinished all the time with no real effect on value because it looks so bad if the finish is not perfect. I think a lot of mid-century stuff is in the same category.

As for glass, it tends to ruin the finish over time because of small amounts of condensation and light refraction on the surface from the glass. Spacers are helpful but not a cure all.

And, besides glass topped furniture looks so "motel".....

Well, sure, but if it's like good flatware where you won't use it unless you can put it in the dishwasher, isn't that better than not using it at all?

Exactly my point!

Except to me the dishwasher equivalent is the glass top.

Just until the kids are bigger...

My kids were raised with wood tables....without glass tops. As a result they know that you don't color on the table top and don't leave your glass on the table, unless it's on a coaster or a map. They learned to put down newspaper if you were going to paint your model there or do a glue and glitter project.
My daughter's kids do the same...but my son's wife grew up in a "plastic cloth on the table" and "foil under the candle sticks" family, so their kitchen table is a mess....mostly with water rings.
The dishwasher won't hurt your silver....and doesn't change the way it looks.
What do the kids do to that table? Set a glass on it? Have some big plastic coasters.....that go to the table when a glass does. Do you eat on the table without a placemat? lay the dishes directly on the table? Learn to use your table not abuse it.
Linda C

There is a lot of difference in durability between delicately veneered tables intended for special occasions and the beefier solid wood ones meant for every day use, though. I've got a cheap solid something I picked up in college that is abuse proof. I've stood on a milk crate on top of it to drywall a ceiling. It's been used as a sewing table and a craft table and a work bench and still cleans up nicely. Then I've got my grandmother's "good" table from the '40s. That thing dents if you look at it funny. It's not just the kids...

Stacy just wants to be able to use her table without it looking bad and without damaging it further. I don't know how to do it, but I certainly understand the impulse!

I think this should be solid teak. I would get it refinished and use placemats--and learn to live with a certain amount of imperfection. My Stickley table came with a little tag that said that it was meant to be used and if it didn't get little scratches and dings in it, we weren't giving it the affection it deserved.

Teak is very durable as it is used on boats! Once you refinsh the top put a table pad on it.

Here is a link that might be useful: table pads

When you put table pads on your table, you are not "using" it....it's being a support for table pads.
As I have said...use it don't abuse it....placemats and coasters....and a good coat of pastewax every 6 months. Wipe up spills promptly, have a caddy for your wine bottle and a small tray or large trivet for the collection of stuff like ketchup, mustard, jelly, syrup, hotsauce etc.