First some terminology and ground work....
matter is a new communication protocol, it specifies how commands are formatted, pretty much anything can support matter with a software update. Basically before device manufacturers would have to add their own native control, and add homekit, Alexa, or google home as separate code for each one. Now they can add matter and be compatible with all 3.
homekit is also a communication protocol.
Thread is a new type of wireless, It's designed with home automation in mind. It lower bandwidth than wifi, but is handles lots of devices well. You need specific radio hardware to use thread. for thread you'll need at least one thread border router (connects to both your home network, and the thread network, and moves data between the 2). Newer AppleTVs and homepods are thread border routers. It allows you to add devices from multiple manufacturers without having to add another box, like you do with hue.
another way to think of homekit and matter is as languages, just like English or Spanish, with their own vocabulary, and ways to format sentences. And just like people and languages, some devices can speak more than one, but as long as the controller and device both speak the same one, they can work together.
Thread is a way to communicate, like a cell phone, or land line. Both do kinda the same thing, but need specific hardware to make it happen.
there's not really anything as a matter gateway.
Thread and matter are 2 separate things, but both were released about the same time and developed with the other in mind.
You can do matter over thread, matter over wifi, homekit over thread.
Homepods and appleTVs, in addition to being thread border routers, are also homekit home hubs (the older models without thread are also home hubs). Which means they function as a centralized point of control for homekit. If your phone isn't at home, any automations will be run by the home hub. The HH will also act as a relay point if you're outside your home, letting you control your home on your phone from anywhere you have network access.
With matter, you can connect to multiple controllers, So your google home and iPhone can both control the same light, both speaking matter directly to the light. Both controllers are not really aware of the other, but if you turn it on with google, the light will tell apple that it's on.
now, on to the question.
windows machines can't control things using homekit or the home app.
BUT.... you can add matter devices to home assistant (currently beta, I think). With home assistant, you should be able to control your devices from the web interface on your PC. Home assistant also acts as a homekit translator, so any device in home assistant can be added to the home app on your apple devices
For me, I use siri much more than the home app to control things, so control from the PC might be less necessary than you think. You can create scenes, and just say "hey siri, gaming mode" which would bring up pre-set levels on your devices. You can Also ask for things by room, or by name. "hey siri, make the living room red" or "turn on the couch lamp" or "turn off all the lights" Home also supports zones, so you could assign rooms to upstairs or downstairs, and then you could also ask siri to "turn off the downstairs lights" Or rooms called "front yard" and "back yard" in a zone called outside.
There are also homekit buttons, so you could put a button by your PC that would bring up a gaming look when you press it, or turn everything off if you long press (most support regular, long, and double press) There's one from Onvis, which has 5 buttons on it, and they all support all 3, so 15 different actions. It connects via Thread. Others connect via Bluetooth, as long as your HomePod/appleTV is in the same room, or maybe through one wall, they might be ok. Thread will be more reliable though.
The least complicated option would probably be to keep it all in home. and just not try to control using your PC. Get a HomePod, so you can add thread devices. Put the HomePod where you think you'll be giving the most siri commands, especially if you don't usually have your phone with you in that room. Go with homekit or matter devices, And since you'll have the HP, that opens up to thread devices.
you can also go with an appleTV over a HomePod, The aTV is a pretty solid and well supported streaming box. You will loose the ability to just shout into the air for things (although your phone might hear you). The aTV is much better at being a streaming box, than the HomePod is at being a smart speaker. You can still use siri on the appleTV to control your house, but the TV must be on, and you have to hold a button on the remote, as opposed to just being able to speak to siri.
If you don't have an Apple Music subscription, the HomePod looses a lot of functionality. You'll be limited to airplay from your phone or PC for music.
