Your failed order with another company has nothing whatsoever to do with BT , if they said the blocked cable / duct was BT responsibility, that pretty much shows they don’t know what they are talking about , as BT have no responsibility for this , it could be Openreach duct , but Openreach and BT are not one and the same organisation.
You haven’t really given enough information about this Alt Net, if this Alternate Network Provider uses Openreach infrastructure via PIA ( Physical Infrastructure Access ) and a part of the infrastructure they wanted to use is the duct under your garden and isn’t serviceable because of a blockage , this Alt Net can install their own duct across your garden from the footpath if they wanted , but many Alt Nets don’t do anything other than straight forward ( cheap ) installs , so won’t consider this.
BT are pretty obvious not liable for OR , OR would clear the duct under your garden, if this Alt Net asked them , and agreed to pay OR to do it, presumably they have not done so.
If you are thinking OR should clear the duct at their own expense , why should they ?, it’s not currently stopping OR service into your property, PIA doesn’t work like that, if they want cheap access to OR infrastructure that isn’t serviceable, they can use various methods to get it made serviceable, or provide their own infrastructure, that they don’t then have to share with anyone else.
You didn’t have to return to BT , there are dozens of providers that could use the same Openreach infrastructure that BT use to deliver your previous and current services, if you felt the time constraints placed on you by this Alt Net ‘failing’ at the last moment, meant that you ‘reluctantly’ had to use BT , how on earth could that be considered BT’s ‘fault’ ?
