I wonder if you spoke to a rock shop. ( lapidary) because if they polish stone and gems they would use a compound that could cut glass or harder stuff ( look up Mohs scale if you aren't familiar with the hardness measurements)
perhaps with something like an auto polisher could polish where the surface is etched.
I think if you use polishes, that will fill the microscopic voids and look better but that would be temporary.
with the touch ups , you can probably use acrylic enamel but you need it to really match and not be two slightly different colors of white, then just use it where needed, use tape to protect where you don't want paint and pull the tape while the paint is wet so it doesn't leave a hard edge. of course it needs to be super clean so I'd use some harsh solvents acetone alcohol, perhaps even an etch. but just on the spot you need to paint. if you want a better paint repair use epoxy , now you need a small quantity and to use the epoxy hardener but epoxy paint will last better than enamel.
If you polish you might risk polishing through , you might need some different grades ( coarse medium fine)
you might use other tool like a dremil for tight spots, but basically a pad or rag cloth is saturated with the compound and run against the surface, perhaps at a very high speed.
a quick search showed this glass polish.
https://www.3m.com/3M/en_US/company-us/ ... 872&rt=rudthere is a common compound called red rouge. it has fine grit but in order to work the grit needs to be harder than the surface you are polishing so, the grit needs to be harder than glass. I'm not sure that is. in order to achieve a polished surface it needs to be fine but you might need different grades and do them in sequence. If the polish is too fine it wont' have much effect without a lot of work. If its too coarse it might be less than perfectly shiny.
If I polish parts that I have made, or if say I wanted a doorknob to gleam and shine, I put a rag wheel on my drill press then hold the polishing stick against it so the wheel picks up the compound, then you polish away on the part. occasionall y strip the compound by holding a chisel to the wheel than recharge it with fresh grit.
like sandpaper there is a sequence of grits. . with the rag wheel you need to be careful because it spins very fast and if things catch they fly out of your hands, but a tub is pretty smooth mostly. In that case of course you can't hold the tub to the wheel so it needs to be a hand held polishing wheel of some sort. perhaps a rag wheel in an electric drill could work , it would be worth trying I think.
You have to be careful not to polish right through, perhaps with a fine grit that would take a lot of work and time but it needs to be appropriate. If you get into the technical side perhaps you'd consider the surface speed, the circumference x RPM will give you that. Some technical info may refer to a recommended surface speed.
It's like auto polishing but car paint is a lot softer than glass so those products probably wont help so much, but the polishes would give some temporary satisfaction.
people who polish rocks and gems will perhaps advise better?
other than catching and flying , be aware that a spinning wheel can throw stuff off in your eyes. rotating tools like grindstones and attachments have a maximum RPM. usually as the diameter increases the speed decreases because whats important is surface speed. with things like grindstones if you spin them faster than recommended they can get going too fast and fly apart. a little dremil holds small tooling and normally they go up to about 20,000 RPM whereas a 6 inch wheel might run at 3600 RPM or something like that. Its usually marked on the wheel when you buy it.
There is also a tool called a die grinder. I have one. it as a 1/4 inch chuck and like a dremil It will hold different tools. Usually with that I will use various cutters that are designed for steel. I can carve steel with it.. so say I had a hole through 1/2" thick steel and I wanted it to be a wide slotted hole I could stick it in the hole and move sideways. they are air powered , super high speed hand held. BOTH HANDS! This is a tool I'm very cautious with because it is so extreme, there is no protection between you and the tool so one slip can easily and instantly make very serious wounds. for that reason I use it carefully and often find other means. Both face shield and safety glasses are mandatory. They aren't that expensive but you really have to be careful if you use them. they are very powerful whereas if you use a little dremil , those spin fast but the tools are much smaller and less aggressive, although a dremil has high speed they aren't so powerful as a die grinder. both have their place. I think to be successful and not take forever you need a tool more powerful than a dremil the motor and the grit do the work. All these variables need to be appropriate.
you might not need a super high surface speed, this is something you could research more. you have several variables, the surface speed, hardness of grit, the wheel may be soft or hard or larger or smaller. the grit needs to be appropriate in hardness and also in fineness. how powerful the tool is, how much pressure you apply, how quickly you move over the surface. I think this might involve either more training from someone experienced with it or some experimentation on something you dont care about.
If you happen to have a contact of a metalworker that does commercial kitchens, they can make custom stainless counter tops and things and polish them. perhaps they would have these skills down.
I think if it were me I 'd just try on an old sink found in an alley and whatever grits and tools I could find, but I would expect to experiment , maybe burn through on my test object and wreck it just to see what happens and learn in order to get the variables sorted out. I think first I'd try a 6 inch rag wheel about an inch thick, in an electric drill with an appropriate grit, see what happens, what's to loose?
my brother was a dental technician , they work with porcelain and polish it. He gave me some wheels for my dremil. they are basically about the size and consistency of a pencil eraser on the end of a pencil but the rubber is impregnated with the grit. they wear as you go. as you get them started you hold them against a hard stone and run them and that makes them balanced , then you can increase the speed once they arent' shaking the tool. That's too small physically but gives you the idea. not a lot of difference between a dentist drill and a dremil but they aren't intended for big surfaces so you need something bigger than that. I think the speeds and type of grit would be similar.
