Birthdays being on the same day of the week is a poor objection: your birthday already falls on a weekday much more often then not, and chances are you celebrate it on the weekend before or after. We'd lose variability in holidays that are "last Thursday of the month": they'd just be on a specific date every year.
One nice thing we'd gain is that the phase of the moon would shift much slower with respect to our calendar (the moon doesn't do a Year Day). I suppose some superstitious healthcare workers might object to this. If you think about it, there are many more things naturally tied to 28 days than 28/30/31.