Perhaps it would be easier to have the government collect all of our income, subtract the taxes, and pay us the difference once a year. That would be even simpler.
If you were to pay £80 into a personal pension (similar to an IRA), about a month later an additional £20 would be added. That's tax relief at the 20% basic rate. (Higher rate payers currently need to file to claim additional relief, but there's no reason that couldn't be automated too.)
If you had a company pension instead (similar to a 401(k)), you can pay in through salary sacrifice, where the salary your tax is calculated on is reduced and the difference paid directly into the pension scheme.
Whichever route is taken, the end result is the same: a £100 contribution cost you £100 in pre-tax money.