>In practice I would probably try other means first to heighten vacc level: ask for tests more often but refuse to offer them for free to unvaccinated, cover only 10-20% of any COVID treatment cost for unvaccinated grownups, as it was their choice to heighten their personal risk by not vaccinating(with some exceptions for those unable to vaccinate for medical reasons - those have to checked in advance). Alternative ways to monetize possible, like a monthly discount for vaccinated people in health insurance cost. More testing/research on Corona infection in kids compared with risks of vaccination would be nice as seen from this keyboard.
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>BUT: If the strategy is to give Corona antibodies to total population (which WILL happen everywhere, just a matter of time as it becomes endemic...) to avoid lockdowns in cold season 2021/2022 at all means, starting in summer is smarter - as long as enough hospital beds are free, this alternative/more traditional way to get antibodies will add a few million before October, which traditionally hikens aerosol-spread infections.

From a public health POV the unvaccinated should be tested more often, and encouraged to get tested. Charging for tests is exactly the wrong thing to do. Reducing testing amongst the unvaccinated will increase COVID amongst them and their contacts, lead to more cases of serious disease and provide more opportunity for viral mutation.