As you identify, Air Italy did have a duty to re-route you under EC261, on other carriers if they could not offer the service themselves. However, given you eventually accepted a refund, that option has now disappeared.
There was also no guarantee that any re-route would have fitted with your positioning flights anyway. The same would have applied if they had continued running the route but with substantially changed departure times.
I think, in the circumstances, I’d probably regard only losing £175 from the BA Avios cancellation as something of a result. I certainly wouldn’t be initiating an Amex chargeback, which - as I understand it - puts a black mark against the merchant even if not upheld. I’ve no idea if BA keeps a record of those sort of things, and what influence it would have in the future on goodwill gestures if they do, but it’s not something I’d personally, on balance, want to take a chance on. You never know when you might genuinely need their discretionary help ...
Every itinerary involves some degree of risk. Leaving from a third country increases it. Every time an extra separate ticket is added in puts it up again. Many of us gamble on substantially cheaper fares that don’t originate in our home country - and we have to accept that if something goes wrong we get a relatively small financial hit compared to the savings we make overall. I know it won’t be much consolation, but you’ve been spectacularly unlucky to get hit so hard because of the expensive TG flights - albeit it no amount of sympathy is going to recover that expenditure.