Given the blatant exploitation by these companies of the 'they're subcontractors, not employees' route, it might not be entirely unreasonable for the government to turn around and point out that they clearly
are employees, and that the liability for ensuring the vehicle used is appropriately insured (as well as taxed, maintained, and the driver adequately licenced) falls upon the employer.
This doesn't entirely solve the problem, as the delivery driver could still turn up in a vehicle other than that agreed with the company. The fix here would require a fourth party (i.e. the restaurant) to validate the vehicle used, but this would probably add undue complication.
The difficulty my wife and I have with 'business' insurance is that we only have one car. Most insurers only cover business use for the policy holder, and not for other named drivers.
Occupation does have an effect on car insurance and, for people whose occupation may fit more than one category, it can be worth playing around with wording to get a better quote. In fairness to Aviva, other than the specified prohibited occupations they do not base their price on occupation.
I have never found an insurer who recognises the existence of my profession. Not all of them even recognise my industry.
A colleague insisted on getting their profession listed 'correctly' when applying for life insurance once, then somehow managed to convince the insurance assessor that it fell into the same risk category as saturation divers and test pilots. This did not do their premiums any favours.
You're lucky. When I joined the railway and changed my employment to "signalman/railway worker" my policy increased slightly. The reason given was I would likely to be driving at odd times of day and possibly tired. Colleagues have confirmed similar.
I don't believe they even have to justify
why your profession impacts your premiums, as seen by the fact that there's always been a high level of claims among people with south-facing garages (or some such nonsense) when it's time to renew your policy.