Does TSA allow travelers to carry a walking stick with metal tip on board plane?

It's a grey area. Technically they are considered to be like ski poles, and have to be checked. But the agent could also consider them to be walking aids and allow them. Also one airport could allow it as a check-in and if you have a connecting flight the next airport might not allow it. These sticks can be pricey so I never take the chance and check them.
Earlier in the year I bought an awesomely beautiful decorative apple peeler in Paris. At CDG for my flight home security wouldn't allow it in my carry-ons and had to throw it out:( So anything can happen.

...stick with metal tip....

It would likely depend on the metal tip. Is the tip is sharp and could be a weapon, it could be prohibited. Or if it is deemed a club. See:
https://www.tsa.gov/travel/security-screening/prohibited-items

Sports equipment that can be used as a bludgeon (such as bats and
clubs) is prohibited....
Even if an item is generally permitted, it may be subject to
additional screening or not allowed through the checkpoint if it
triggers an alarm during the screening process, appears to have been
tampered with, or poses other security concerns. The final decision
rests with TSA on whether to allow any items on the plane.

Being mobility impaired can be an advantage. My wife strained a knee in the UK. She was offered a wheelchair at Heathrow and we were speed through lines. We made better time than had she been injury free.

"Walking Stick" is a broad range of items.

If you are referring to a cane that is needed to walk from point A to point B and obviously looks like one, you won't have a problem. As states earlier, it will get screened seprately but it should be allowed.

If you are referring to a long pole use mostly for hiking with a partially sharp end point then it is prohibited.

A few years ago I sprained my knee the night before a flight and had to use a folding cane while going through security. TSA couldn't have been nicer. I had a knee brace on and they even helped me get my shoes off and on. But it obvioiusly looked like a cane.

I'm old enough to have watched the Bat Masterson TV show.
http://www.fashionablecanes.com/80485bat.html

.television series, which aired on NBC in 108 episodes from 1958 to
1961 and featured Masterson as a superbly dressed gambler, generally
outfitted in a black suit and derby hat, who was more inclined to
"bat" crooks over the head with his gold-knobbed cane than shoot them.
This cane replica that is available should not be used to bat people
over the heads, if you can manage to control your urges.

If it looks like a duck....

Vicki - TSA offers assistance through their Passenger Support services https://www.tsa.gov/travel/passenger-support - I'd suggest contacting them directly regarding your "mobility aid".

I routinely take a collapsible hiking pole as a walking aid. Strapped to outside of carryon, and never questioned. Its a cheap one, so that if I had to surrender it and get a replacement on the trip, it's no great loss.

Believe it or not, you can buy (in the US) full-size canes/walking sticks with concealed stilettos in the handle, so that its not unusual that TSA would give them closer scrutiny.

Walking stick that acts like a cane is just fine.
Walking sticks that are "hiking poles" are not. I carried mine throughout Europe with no difficulty but was not allowed through security in the US with them as a carry on. I had to check them at the last minute or lose them. They are considered a possible weapon by TSA.

I have traveled with a cane after knee surgery and there was no problem whatsoever.

I had no issues carrying mine from ORD to AMS in my hand at the airport or thru security. Then from FCO to ORD same thing. They didn't even look at it.

I faced this issue last fall. With problematic knees and getting ready for a trip that would involve somewhat rugged terrain in Greece, I bought a folding pole-like stick. Because of the issues mentioned, and because Greece doesn't allow metal tipped canes or sticks around their ruins, and because there are no handrails, I got several rubber tips for my one pole.

My orthopedist wrote a note to put in the bag with the folded stick. I went to the Tucson airport and discussed this with the TSA supervisor there. She said I would have no problem leaving it folded in its bag in my carry-on, or in using it through the airport. I got her name, just in case I might have any issues with anyone else.

Why didn't I get a cane? Because I'm tall and most canes are too short for me. I don't need something to lean on (yet) while I'm walking. I need something to steady myself and maintain my balance.

I don't know if this factored into the situation or not, but for all the trips I've made over the past couple of years, I have been give Precheck status. Didn't ask for it. Didn't pay for it. It just comes up when I get a ticket or a voucher to fly, no matter what airline I take. Hmmm? I hope I didn't just jinx that...

I only used the pole once, for the Acropolis, and I definitely needed it there. I should have used it at Delphi, where I slipped on some loose dirt and landed on my keister. I checked the bag with the pole in it as needed on the intra-European flights and carried it on for the US ones and the ones over the Atlantic. Nobody ever questioned it.

As with all things involving any kind of rules, YMMV.