There are plenty of firms that claim to 'specialise' in particular aspects of motor insurance but are actually just general insurers using a particular trading name to attract a particular type of customer.
For example, Sheila's Wheels seeks to attract women drivers but it's actually just part of eSure, that insures everybody anyway.
It's just the same when seeking insurers for drivers with penalty points on their licences. Many (most?) of the 'specialist' insurers actually turn out to be general insurers simply hiding behind a different brand name. (That might well have been the case with the ad that you saw).
While Adrian Flux is also a general insurer, they are primarily known for insuring drivers with particular problems (such as having convictions or, for example, driving big American cars), so they might well be worth a try:
https://www.adrianflux.co.uk/convicted/