We require vaccinations for schooling and for healthcare already (e.g TB). This is no huge stretch.
Think about it: two people enter a workplace. One is unvaccinated but recently tested, so it can be said with good confidence they are not infected. The other is vaccinated but may well be sick and spreading it, especially as effectiveness wanes quickly. The other is waved through, the first is penalized with constant charges. Who is making other people the safest here? It's not the vaccinated guy.
It's all about risk. Someone who has been vaccinated has a very low likelihood of being sick with Covid. Someone who has recently been tested also has a low likelihood.
IMO, though, if the tests are a week apart, that's a lot of opportunity for Covid to slip through. There's a reason that 72 hours (and sometimes 48 hours, I think) is the time limit for tests for traveling. Even that isn't a guarantee, though.
I can't find the chart now, but Covid progresses pretty rapidly, so if someone gets tested on Friday and was exposed 1-2 days before that, they can pretty much spread it for an entire week before they get tested on the next Friday.
And even if they were tested on Mondays, they could still be spreading it before the weekend.
Tests prevent someone from spreading it for weeks, but not days.
Vaccines are much more likely to prevent the spread at all.
You can go back and look at 10+ years of UK NHS "hospitals desperately overwhelmed" annual articles. Just because you hadn't heard of it before doesn't mean it isn't normal.
In the US, they've relaxed many guidelines and allowed student nurses without gaining licensure yet to practice because of the need [1].
[1] https://www.healthaffairs.org/do/10.1377/hblog20200327.71403...
Mumps
Smallpox
SARS1
MERS
Chickenpox
Tuberculosis
To name a few.
Not a fools errand. Important to get vaccinated to prevent worse variants.