Because he isn't having clients coming and going from the house his insurance could potentially be covered under house insurance, especially with covid. Since essentially he is doing admin/clerical work (no physical activities and no equiptment). That wouldn't be uncommon and is an easy qdd on these days for very little extra premium. That would cover his PL insurance for business use.
EL cover isn't required for himself, or any director (possibly roped Sam in on this promising dividends/shares....eventually)
EL is also not required for subcontractors - so cash in hand jobs where you should be paying your own NI and tax won't need it either. Kaz and Sean could both be on these types of agreements. As long as you work under your own self esteem. Though a policy for the type of "work" he does would be circa £150, if that, so I wouldn't say it's impossible for him to have.
As for the walks, unless they are a paid for service he has no real liability there either. He can encourage you to go to a public space and if you all go and have an accident or moment of ill health, that would not be classed as his fault. If you are paying him to train you (which I don't think DJ was) he has no liability, beyond having a moral compass to do what's in your best interest as a fellow human. No different to a friend getting you to go to the park really. You'd have to prove it was his fault you tripped and broke your leg while there, due to his negligence in his expertise to claim anything from him. Not just his fault you were in the park, if that makes sense.
Again, a policy for him for that type of work wouldn't be hugely expensive, so it's not necessarily something he wouldn't have. Especially when he was taking off like he was last year, he might have just used a couple of hundred quid to cover himself.... well maybe unlikely but you never know