They do. It's called the 61.75 license, and is readily issued to any ICAO license holder.
Your concept of "readily" and my own are clearly at variance.
If an FAA license holder wishes to fly G Reg in UK airspace he can, with no formality.
If a UK license holder wishes to fly N Reg in US airspace he fills in
this form and sends it with a copy of his license and medical. Because he's from the UK then he first needs to fill in form SRG1160 from our own wonderful CAA and pay them a fee (I believe it's 40 quid) so that they can send verificatiion of his license to the FAA. A similar arrangement is required if he has an Australian license (why are the Brits and Australians singled out for this treatment?)
Having done all that he has to present himself in person to a FSDO in the USA or to a DPE (and pay him a fee) from one of the Field Offices at some pre-arranged future date.
The UK recognition of the FAA license and the FAA 61.75 process (and I have one) are not quite equivalent n'est ce pas?
Oh, and by the way our cousin from across the pond flies on his own license with his own ratings and does not have to do our bienniel hour with an instructor but the 61.75 holder doesn't and has to do the US BFR.
I think I have that right but am happy to be corrected.