Some people go into university straight from S5, so your father isn't wrong in that regard. But as above you do need to have already met the course requirements by the end of S5 for this to be possible. If your school is telling you that you can't go to university straight from S5 then it's likely because you haven't met the requirements yet.
I personally think most young people benefit from that extra year of school. Taking more Highers broadens the range of subjects you can pursue in university and taking Advanced Highers will make your first year at university easier as you'll be working at a similar level already. Being under 18 at university can hinder your involvement in the social aspect as well.
Your dad can't decide for you whether it's best to leave at the end of S5 or S6. You need to make the best call for yourself and he'll just need to accept that.