I don't understand. Why isn't it standard practice to just issue a set of "personal days" without regard to "sickness", "leisure", or "mental sanity"? My company switched to this a couple of years ago and doesn't require an explanation. They are your days off, take them as you see fit.
It's strange to have that allowance laid out as a fixed number up front.
Most of my jobs in my career have offered 25-30 days paid holiday, plus the 8 or so days of public holidays, and paid sick leave.
That way you don't fall into a trap of debating whether you should use your last allotted "personal day" on something you might thing isn't worth taking it for as you might need it later.
