With so many unfilled positions - in restaurants, retail, construction, etc. - is yet another government program really the answer? There's plenty of opportunity in the free market to be had. This reads like an ode to a more romantic time, as misguided as even that might be.
Basic minimum: prorated benefits. Pick a cutoff point for full time (say 20 hours) and every hour for a worker under that is pro-rated at their fraction out of "FULLTIME" hours a week for the pay period.
Taxes == Benefits: Government does PTO, medical, etc; make everything EASIER FOR BUSINESSES, no more worrying about these things other than over the tops they aren't required to provide. Make those easy too for Paid time off and any other multiplicate benefit a flat multiplier if they want N% the standard rate.
The biggest sticking point for some workers revolves around time. Be that work-life balance (partly fixed by some the above), unpredictable hours, or hours that no one wants. It needs to be easier for the workers to have a __viable career__ as a whatever, not just be a cog in a machine that is summoned like a slave at the business' behest.
Work/Life balance might include kids or anything else outside of work that's important. Where it doesn't really matter there's an obvious answer. Where it does, like public service, maybe they need workers with careers covering that slot; who expect to regularly work that shift on a predictable schedule.
There should also be a formally allowed process for schedules. If it's specified in the primary worker contract as a fixed base schedule that's one thing. If it isn't then the employee should be able to decline, as a contractor that refuses the offered hours, with no other detriment (obviously non-take home pay, but no other penalty).