@Whoknowswhocares
As a trainer if I said someone needed a set number of lessons, it would be that I needed that amount of time to teach the owner the techniques that they need to use in order to gradually improve the dog, NOT that there was a specific timeframe or expectation on the dogs improvement.
If I need 6 sessions to build the skill set of the owner and get them started, it is highly unfair for me to continue to charge weekly to go over the same ground.....I would expect the owner to go away and practise what they have learnt, be in contact with me in between whenever they wanted it and then to do a follow up as necessary.
Perhaps your trainer meant something similar and has just worded it badly?
This x100
Have you been following the trainers' advice, in full, consistently and without fail? Dogs thrive on consistency, and by far the biggest challenge trainers and behaviourists face is getting owners to follow their advice. No reputable trainer will guarantee results.
You should also, however, be realistic about what is achievable, both due to past trauma (rejection by mum, for instance), breeds (both poms and chis are notoriously yappy) and so on.
Positive reinforcement training does work (and you say you've been seeing results), but it takes time, effort and consistency.
I do understand the difficulties in having a dog that bites and is tricky to walk. DDog (rescue) used to bite if you touched his paws, bite my lower legs if he saw a motorbike (he drew blood on many occasions), bark wildly if he saw a dog he couldn't go and say hello to, hates specific breeds and so on. We've come a long, long way using positive reinforcement methods, but I can assure you that it took a lot longer than six weeks and he'll always be a work in progress (I've spent two years feeding him a treat every time he sees a motorbike, and that's unlikely to change any time soon, though I no longer have to dive into stranger's front gardens to avoid them).