There would be very easy means of reducing population size (and probably much quicker ways) that wouldn't involve giving individual choices. They could decide to stop treating a lot of conditions, just not fund healthcare for various ailments. They could cut ambulance numbers and reduce ICU beds further. We could stop blood donation drives. There's loads of options.
They could piss off Russia, China, North Korea, etc. Ignore extremist threats, etc.
They could have not bothered with lockdowns and isolation measures and made a decision at the beginning that Covid cases weren't going to hospital for treatment, etc. They could have made sure that positive cases were being sent into every residential facility.
The thought process behind the 'population cull' honestly baffles me as let's be honest, if they wanted to, there are so many ways to do it which isn't going to be as protracted and unpredictable as trying to get everyone to agree to vaccination. Plus, why would the government be targeting everyone at the risk of leaving these deluded arseholes behind!