Room correction in noisy environment (ARC, Dirac, Minidsp, REW)

Sounds like you already have a DSP unit. If not, you can consider the Linn Selekt DSM which does not use a microphone, instead it uses math.

https://www.linn.co.uk/technology/space-optimisation

https://www.soundstageaustralia.com/index.php/reviews/282-linn-selekt-dsm-katalyst-network-music-pla...

The Linn's DSP is considered comparable to the Lyngdorf Room Perfect system.  The Linn is only Roon Compatible and not Roon Ready.

I don't understand how a system can offer room correction without using a microphone and sending actual sound waves out into the room, analyzing it, and adjusting for the response.

The first link I provided explains Linn's rational. Linn actually thinks along the lines, how can you do this properly with a microphone given the problems associated with that, such as the background noise you face. Linn has a long diatribe against the microphone in DSP setup  on their web site (too busy now to find it).

I am going to demo a speaker with too much bass for my small treated room. So I revisited some DSP systems that I looked at before. The Linn SPACE OPTIMIZATION is considered very good. The Lyngdorf ROOM PERFECT system is I believe older and maybe benefitting from that. I am going to demo with the Lyngdorf 3400 because I want ROON READY. I think both products are likely similar in quality, but they use completely different approaches. At the dealership that carries both, along with the Athem STR preamp with ARC, one guy marginally preferred the Lyngdorf and the other marginally preferred the Linn. 

First chance you get, go on-line, look up Fletcher-Munson curves. Then do a search for room reinforcement, nodes, frequency and speaker placement. If you're smart it should take like maybe an hour to understand that no matter how quiet you can get your room, the best you can hope for with DSP is to get it almost as good as you could do by ear. And then only at one volume level. Even then only in the one location. And only at the cost of making things worse everywhere else.

In other words DSP is a colossal waste of time and money.

That's the good news. The better news, hardly anyone bothers to do the reading and thinking to figure this out. There's more technically unsavvy audiophiles than you can shake a stick at. So there's a ready market and you will have no trouble selling your DSP to one of these rubes.