In my long experience its not at all possible for BT to say two things at once and consistency in ways and patterns of installing is patchy to say the least. Have you ever read their small print for letting an engineer into your home? I have, We had some complex work to be done so I sent it to my insurers and they refused to accept the liabilities imposed. I ended up with a local contractor instead.
There are two issues here.
The first is to ask did BT make any kind of quotation verbally or in writing based on their 'testing' of the line and before you made a decision instructing them to re-connect? If they did were there any conditional statements or provisos . . . eg. subject to this that or the other? If they did 'quote' rather than tentatively 'estimate' hedged about with conditions then you can hold them to that quote. After all you were entitled to rely upon their expert opinion. Otherwise your'e a little stuffed!
The second is that for reasons of speed and economy cables usually run in straight lines. There is nothing to stop you digging your own exploratory trench to find the cable. If you do not find it or it is severed you can then ask for a quotation to connect in your own ready made trench from boundary to house.
Indeed, if it is not too much hassle for you there is no reason why you cannot have an outdoor termination in a sealable chamber at the edge of your property. You can have the master or primary socket installed there and what you do after that so long as it meets regulations and any planning constraints- you probably can't put up a pole and have slung conductors to the house, garage and garden shed! - is entirely your own business. One of our family homes in a deeply rural area has this, it saved a fortune in BT sending contractors to trench everything in from the boundary to the house. Do not be put off by spurious claims that you are not allowed to do that. It may be unusual but its not impossible.
And, yes, there is an element of rivalry by sabotage we encountered it ourselves when we bought a little property for temporarily housing a student. In this instance it was the phone installation that did for the cable tail.
Anyway, hope things get sorted for you.