YANBU
He may fee, he is supporting/doing the best by his son, but he isn't. He is encouraging him to spend all day not doing anything to support or better himself. That isn't the best course of action for his older son, or his younger children. If he was studying in some way, or trying to support himself, and needed something extra then maybe it would be different, but even then the amount is way more than it should be.
If he is not working he needs to make a claim for benefits. My guess is that he doesn't want to because
-dad is bankrolling him
-claiming benefits means that (unless medically he is unable to), he will be expected to apply for jobs, work on his cv etc.
-if he doesn't have any qualifications (you indicated he hadn't finished school) he may be expected to get some form of qualification. Without it, he is unlikely to get a job, unless he can show an employer that he will put some effort in, and is worth taking a chance on. Doesn't sound as if that will happen - he will way down the list in terms of candidates I would pick to employ.
On a side note - is the money that is being received by the son/mother being declared in relation to any benefit that is being claimed - my guess would that it isn't.
As someone else has said, if he isn't claiming, he won't be building up credits towards a pension. However, I doubt that is of concern to the son - he will just expect to be supported anyway.
Lots of people have to claim because circumstances change without warning. Assuming he gives him the money at the end/beginning of the month, I think he should say that the payment at the end of Feb/start of March is the last full payment. The son needs to sign on/put in a claim for benefits. Maybe allow a month at half the amount during the transition until benefit comes through, but after that he is on his own.
However, maybe he could offer support if needed for job hunting, eg new shoes/clothes etc for an interview.
If he feels guilty not supporting him, maybe be could put some money aside into a savings account for him. The son/his mum do not need to know, but it means his Dad has something there if the money is needed for a specific issue eg he gets a job and needs to buy tools etc.
If he does put money aside for his older son, he should also be putting a similar amount aside for his younger children. £100 per child per month would start building up nicely, but would be significantly less than he is paying out now, and would make a massive difference to the lifestyle you are able to give your younger children. £200 per child per month would still only be half what he is paying now, but each child would have £2400 at the end of the year.