I don't live in the UK but private liability insurance isn't a thing where I live either (Ireland). I already pay for life insurance, house insurance (includes contents cover), motor insurance and mortgage payment protection. That's a lot of insurance.
I had a quick glance at the examples you listed in your post
Tipsy guy on an e-scooter knocked me over on the pavement, I broke my arm, needed an operation, couldn’t work for a few weeks and lost a fair amount of money (self-employed). The other party’s insurance covered my loss of income.
If this happened to me, I would claim sick pay from my employer/the state and my mortgage payment protection would probably cover my mortgage payment(s). Since you're self-employed, I'm assuming you would have some form of insurance to cover you in the event that you were unable to work due to accident or injury. In any case, it's very unlikely the e-scooter guy would wait around long enough to pass on their details.
Someone knocked a brand new iPhone out of my hand in a train station. Their insurance covered the damage.
I'm not an iPhone person and my phone is worth considerably less than an iPhone. So I would just shrug my shoulders and buy another phone. If I owned a very expensive iPhone, I'd either get phone insurance or see if I could get it covered on my house insurance (contents).
In a bar a woman spilled red wine on my quite expensive beige suede bag.
You'd want to be really stupid to bring an expensive suede bag to a bar. In my youth I had a suede jacket. I got it dry cleaned once and it cost me a small fortune. I vowed there and then that I'd never buy anything suede again.
My neighbour’s teen son scratched our parked car with his bike. Neighbour’s insurance paid out, didn’t have to go through my own car insurance affecting my premiums for years to come
I'd consider claiming off my car insurance but it might be cheaper in the long run to pay for it out of my own pocket or save up for it. It simply wouldn't enter my head to claim off a teenager (or their family) when the damage was accidental.
I stumbled over someone’s (badly placed) laptop cable at the uni library and pulled down and damaged their pretty high-spec laptop
I doubt very much the uni would sue you. They would be hoping that you wouldn't sue them due to a accident caused by a Health & Safety lapse.
Friend sat on our white sofa in new coated black jeans, colour rubbed off / bled into the fabric and couldn’t be removed (sounds like this one might be covered by many people’s contents insurance in the UK?)
See comment above about the stupidity of owning a suede jacket.