Rgb to cmyk and colour matching?

OK so I read advice saying it’s good to use png uploads because of using transparent backgrounds, no problem there. But pngs are in rgb so I’ve also read that spreadshirt then have to convert to cmyk and potentially adjust them. I’m just wondering if those adjustments get the printed colours anywhere close to what we see on screen? I understand the difference between the 2 and that colour shifts are likely, but by how much after adjustment? I read advice somewhere on here that basically said the brighter and more garish it looks on screen the more likely it is to get close to as you want it in print. One of my designs uploaded here has been done with very bright cyan writing, I sent it off to another printer (purely for test purposes because they said they could do it quicker and would colour match) and it’s come back almost more grey than blue and just not close at all! I’ve read all sorts that say it’s the brightest colours that tend to be more of a problem in cmyk but I honestly wasn’t expecting that much of one after they said they’d colour match. Is that something that’s likely to be an issue for spreadshirt or will the adjustments do a good job?

Also do conversions work? Like if I find a colour I want in cmyk and then find the approximate values for rbg or even a hex code (I know they might not be exact in a lot of cases) will that make the printed outcome better?
I only have basics not any specialist apps at the moment, I’m very much an amateur and still learning a lot (this is currently only a hobby really) so any help would be very much appreciated.