The original reason for vaccine passports was that vaccines stopped transmission, hence the ability to show a negative lateral flow test instead.
The reason you have given means moving the goalposts to infringe someone’s civil liberties, their freedom of movement, just incase they happen to get ill and need medical care. That’s a very dangerous precedent to allow any government to embark on.
But even then the logic doesn’t stack up.
An unvaccinated person under 40 has a minuscule chance of needing hospital care resulting from a covid infection.
https://www.cdc.gov/coronavirus/2019...th-by-age.html
The same is true for an unvaccinated person who has previously recovered from covid so how does banning these people help to reduce pressure on the NHS?