We had our 17yo PTS the same day his kidney failure made his life unenjoyable (I was in charge of the decision and I said things like if he stopped purring when stroked, started hiding, stopped his daily patrol, didn’t want to eat, had a significant routine - one day he refused food and wouldn’t purr, it literally happened overnight, and that was his last day). I agree with PPs that your DCat should see the vet about any health issues he may already have going on, which will also have the advantage of them being aware you might need them to PTS.
Re your children, although pet bereavements can be as significant as human ones I would say they should go to school as normal, but have some chats with them now to prepare them. Maybe ask them if they want to see the body of the cat, if they want to have a little funeral etc. If they know the cat is going to leave them for a better or at least more peaceful place and that he’s probably old and tired and maybe hurting, it’ll be easier to accept than a sudden loss. From personal experience, I went to work one day and then found out my mother had had my horse shot (I knew PTS was on the cards but I didn’t get a chance to say goodbye). I had no closure and still feel guilty to this day. The cat was a different matter - we buried him and had a little funeral and I was in charge of his body and wrapped him in one of my scarves etc - it’s REALLY unpleasant dealing with a dead body but I got over his death and grieving it so much quicker. We’ve had other cats who have died and I’ve seen and touched their dead bodies and it’s actually helped.
As PPs have said above, if you do need to PTS, too early is better than too late. The procedure is not traumatic at all, the worst bit is getting a tiny needle in their vein but you’ll be given a chance to say goodbye, the vet will be compassionate, and it’s seconds from when the drug goes in to them peacefully going to sleep. Horribly sad to lose a friend but it’s a good way to go, frankly.
Final note: nobody, either human or animal, actually dies of old age alone - it’s always a heart, brain or [insert other organ] condition which become more common and severe as we age. Things like kidney disease, which is particularly common in older cats, do start causing discomfort. For that reason I really would agree that a checkup is in order in case they need a change of food, painkillers, other treatment.