I would suggest the Sennheiser HD 65 TV, I have this thing for years. I have a microphone on the side because the headset has none. But it has impeccable sound.

How much do you want to spend?

Do you want just headphones (with a seperate mic) or do you need a headset with one built in, headphones will sound better for the money, but, you'll need to get a mic (10 bucks off amazon would work).

Finally, do you want closed or open back headphones? Open give a bigger sound stage, but less bass and no sound isolation, closed will generally have more bass better to amazing sound isolation but a less 'airy' sound stage, most (all?) headsets are closed back and if you use the mic a lot, closed tends to be betterso your mic won't pick stuff up.

senheiser is good but no longer available for me since they got taken over unless i want to pay twice the price due to import costs

the one i would recommend for budget is the philips fidelio x2hr most of its build is metal it has a detachable cable it is a open back headphone but even though that normally effects the bass these even beat a lot of closed backs in bass i know because it took me a while to find closed backs since i want both

i also now have the beyerdynamic dt 700 pro bass is great better then the philips which you should expect from a closed back but they are known to be very harsh on mistakes in mixing of music which some people just do not like but for me that is perfect

as for build quality they look worse then the philips fidelio x2hr but they are fine the cable you mention is not proprietary they are just xlr cables and you can purchase any xlr cable and be fine this is the standard for professional audio work which wel is what they are made for studio purposes the philips ones are entry studio headphones

but yeah they come with a price tag not horrible and the sound is great if you do not mind the harshness i mentioned you can literally hear everything clearly separated from each other

there is a alternative but the build quality just aint great while the bass is rated as one of the best which is the FIIO FT1 but yeah its questionable how long they last which heck they should have made it 200 euro and improved the build quality and i would have went for that one over the beyerdynamic

there is 1 note though about the beyerdynamic dt 700 they are build to handle higher volumes so it takes way higher volumes before they distort they are also louder out of the box compared to many other headphones at the same ohm they are really easy to drive

I've had a pair of fiio ft1 from the release day and they have been great in both sound and build quality so far.

I was actually so impressed I ordered the fiio ft7 planar magnetics which are simply breathtaking (but outside of OP's budget sadly) and have their new k17 dac / amp on the way.

HIFIMAN has some great offerings too and both hifiman and fiio can be gotten notably cheaper if you buy them through aliexpress and don't mind waiting on delivery, I saved £350 on the ft7's and nearly £300 on the k17 for example.

You might not like this suggestion but here goes anyway.
I suggest getting cheap but good IEMs. There are a few good ones around now for
a much cheaper price then what equally priced headphones can give in sound quality.

I have a Audeze maxwell headset $300.00 USD here in OZ they they were $500 and they sound very good, a little heavy on the head but sound great but I still prefer to use IEMs that cost under $100 If I am using them for music and gaming without chat.

The huge downside of these is if you plan on using a microphone too,
I just hate speaking with the IEMs in my ears.