If you give everyone 3 hours per day, most people will still try to play at 7-10 pm. There will still be a massive queue and most people won't be able to play at prime time.

If you give everyone 3 hours per day and distribute them to random hours of the day, some people will still get to play more. One person can play at 8 am, 1 am and 3 pm which he got assigned to, while another person will not be able to play at those times and is locked out for 3 days.

Stop trying to punish everyone because you can only play in the evenings.