Most people have actually been saying that you don't need such a ticket, but it's the easiest way to demonstrate that (in theory at least) you are competent to do so. After all, according to the bodies that do such tickets, the holder must have demonstrated some competence at some point.
Someone without any ticket could well be completely competent - but there is no easy way of telling (and "I've fitted thousands" doesn't count as evidence).
Yes, and as you've now found out, they were right.
Well now you mention "Part P", then that proves it since (AFAIK) there is no such thing as a "Part P" ticket. "Defined scopes" membership of an approves organisation (aka 5 day wonders) to allow you to self notify work, but "Part P" is merely the bit of the building regs that says electrical work now has to be safe.
