Who pays travel insurance for business travel abroad? For most people this is only a few pounds but if you are going to the United States and have serious health problems, that might cost over £1,000, if cover is available at all. What happens if the insurance company doesn't pay out? Does the employer pay for treatment while abroad or do they expect the employee to go bankrupt?
Assuming the business traveller is an employee, this would always be provided by the employer. How it's done varies, in big firms there's normally a corporate policy, whereas in smaller ones it's often a case of "find a policy you're happy with the cover on and put it on your expenses", I've had both.
It is simply a travel expense. You'd not buy your own flights or hotel for a business trip, so you'd not pay for your own insurance either. If my employer refused to pay for my flights I'd not be forking out myself, I'd just refuse to go, same with insurance.
The insurance will pay out if its terms and conditions are adhered to. A sensible employee reads these, seeking advice from HR as needed, and if they can't comply (e.g. due to a pre-existing condition) flags them up to the employer, and if the employer doesn't resolve the matter then they don't travel.
Obviously if self-employed you are the company and are responsible for yourself, though if the contract involved billing expenses to the contracting company then it's just another expense to bill.