Is there any such thing or do you really have to spend over £50? Yesterday I hooked up one of my dormant rigs to the TV and fancy some couch gaming.
There are cheaper wireless XInput controllers, but unfortunately they are junk and you'll just end up frustrated with them and buying the official XBone controller anyway.
Nothing except the price. To buy one wireless with the batteries etc is just so expensive. I have two, but sadly they are both wired. The PC is about 15ft away, and IIRC if I used a USB extender it started to misbehave.
PS3 controllers work perfectly and natively on a steam link, but a quick google says you need to manually install a driver to get them to work on Win 10 over bluetooth. They're good, reliable and cheap though! https://github.com/nefarius/ScpToolkit/releases £15(ish) new.
Thought they were less than £30 when I got mine, certainly less than the £50 you quoted in the OP. My experience of PS3 controllers is I never got them working, but others have so could be worth looking at.