Hi iron lady!
I can't remember the exact calculation you need use to take into account how much you will have to pay in
NI, but in response to your first point, the reason you pay it is because your brolly will charge you a certain amount for their services, say thirty quid a week. However, your employer's contributions are going to be a lot more than this.
If the thirty quid is all their making how would they afford to run their business if they were paying this as well?
The alternative for them would be to advertise their services for a fee that includes a percentage of your daily or hourly rate, but I imagine this would both be horrendous to explain/market as well as commercial suicide if no-one else operates like that.
Hope that helps.