To think that, as a manager, the occasional “sick day” when someone needs a break is fine?

As a one off, the principle is kind (if its your business and money and clients) but really it indicates that there is a problem in that your team need to be proactively encouraged to book regular annual leave so they don't get to the stage that they basically collapse in a heap on a Wednesday morning and phone in sick. It doesn't help manage their mental health in the long term.

Why can't they use annual leave?

I think it's dishonest and poor management. Why should the rest of the team be left to pick up their workload because they don't feel like coming in?

If they are ill (and that includes mental ill health) they should take it as a sick day.

You're also creating the expectation that people are 'allowed' to have random lie-ins and last minute days off. This may get you into trouble further down the line if you ever have to do a performance improvement plan. It sets a precedent and practice as well as policy is used by employment lawyers in tribunals.

EDIT:
Finally you are risking your own income and professional development by turning a blind eye to this practice. If you have a 'relaxed' approach to the management responsibilities you are paid for, then your managers may also have a 'relaxed' view about whether they want to support your progress.

I don't care too much about this but I try to create an environment where it wouldn't be required. My team members would tell us when they are struggling mostly and I have been known to send people home and tell them to make up the time another time.