They're not wealthy, no. They're very comfortable. Wealthy would mean that they own all the homes outright and certainly wouldn't be worried over spending too much on dessert. As far as their own home goes, they might own it outright and it's a very nice position to be in, but they still need it to live in, it's not as if they could sell it to use as an income.
Their holiday home will need maintenance and upkeep etc and depending on what it is there may be large service charges to pay. Sounds like they're letting it out if they go on other holidays, perhaps they're using the income from the holiday lets to go on their own holidays elsewhere? Both buy to lets will also need maintenance, upkeep, letting agency fees etc.
Private school costs a bomb. Plus all the school trips and other activities that the kids are expected to go on. Presumably the income from the buy to lets will go some way to covering the school costs, so in effect it's the same as if the kids were at state school with with neither of the buy to let income, same as the rest of us.
So what they're left with is a house they use as the roof over their head and a holiday home which they possibly let out and use the income to go on holiday with. If they're going abroad 3k doesn't go far on a family holiday these days during the school holidays. I have no idea what kind of salary he earns but presumably it is not enough to not worry about cash flow from the sound of it as everything else seems tied up in assets.
They're wealthy on paper, but unless they sell a couple of assets or switch the kids from private to state school they're not wealthy in everyday spending terms.