Yes it does help a lot. No pictures, but I’ve done it twice over a 10 year period. The company that did mine acid washed all of the tiles and grout, then removed grout and regrouted. For a medium sized bathroom (floor and the whole shower) it was about $350-400, so costly, but it looked MUCH better. Like new really.

^^ That's different than just grout cleaning. I used a grout cleaning company a few years ago and they did a cr*ppy job, it was not worth what I paid. I had to do it again myself - hands and knees, scrubbing with a grout brush and paste of OxiClean and hot water --> that worked really well but was a lot of work -- I had a really big area, though (large kitchen, dining area, and foyer). In your size space, I'd probably just clean it myself, unless you want it regrouted.

Yes, it does help dramatically in situations I have seen. The tile pros have very large rotary machines, like carpet cleaning machines, but specialized for tile. Some have a special machine with a narrow brush that goes back and forth on just the grout, like a gigantic mechanical toothbrush. The chemicals, detergents, and hot dirty water are sucked up by the machines, followed by rinsing thoroughly. They can seal the tile and grout again, I highly recommend it for easier cleaning afterward.

A large kitchen and three bathrooms cost a few hundred dollars in a low cost of living area. It took all day. BTW, there are special hand-held electric scrubbers they use for wall and shower tile.

The guy I used didn't use any special equipment, guess it was just a lousy choice of a company by me. Wish I would have known about all this when I needed to!

I hired a floor cleaning company to clean the grout on the floor in my DM's condo - it was stained similarly to the OP's. Made no difference at all. It must all depend on the company & whether they have specialized equipment (and which equipment).

We had a company come out to clean ours and I was hugely disappointed for the cost. I had better luck cleaning them on my own. A pain and time consuming but it got clean. I got one of those brush attachments for a drill for the next time I really clean it. (I wish I hadn’t done tile in my kitchen)
Helpful article: https://brendid.com/ultimate-guide-to-cleaning-grout-and-tile/

I’m a GC We found sometimes jobs like yours
I have a better way then be removing the Existing grout
Is a new product we use come in 8 different colors

Custom Building Products Polyblend #382 Bone 8 fl. oz. Grout Renew Colorant