This is exactly why we should adopt a 13 month calendar, with a Year Day as the last day of the year. If it's a leap year, there'd be two consecutive Year Days. These days should be special: they are holidays, so everyone has them off. This immediately solves the payroll problem for most people: just don't pay them for this day (obviously adjusting the numbers such that if you make $60,000/year, you still get that).
For those who work on this day (medical professionals, etc.) the rate is different (being a holiday and all), so they just get paid by the hour, regardless of what they do throughout the year.
Your payroll can now be weekly, biweekly, monthly, or whatever you want: you still get the same amount of money per month. This is nice because you don't get weirdness with things like rent/mortgage payments which are monthly. For example, for those who get paid biweekly, you get two months a year where you get an extra payment. This is silly and leads to more difficult accounting and budgeting.
Monthly is nice, but lots of your expenses are likely one a week-by-week basis: groceries, entertainment, etc.
Weekly is AFAIAC the nicest: it is the easiest to budget because it's as close to the common denominator as possible. However, it is not the LCD, so it still has the weird problem of getting more money in some months vs others.