Hmmm, tough call this one BBNovice.
The bottom line is, you have purchased a house and all that comes with it. It's neither your fault the wiring is sub-standard, nor Openreaches. Its like finding your electrical wiring was amiss and expecting your local electrical board to correct it.
The problem lies with the fact that the incoming line meets all the extention wiring in this 'Heath Robinson' type junction box. When you say it looks like an "alarm connecting block", do you mean as in one of those magnetic contacts you see on doors ? The reason I ask is how 'ex' is the 'ex-BT engineer' ?? As junction boxes have changed over the years and he may not be aware of the newer types. Long shot, but it's worth asking.
My honest opinion is, ask for a SFI Broadband engineering visit. The cost of this is picked up by the SP, and they decide if this cost will passed on after reading our (the engineers) report once the job is completed. I can't speak on behalf of the SP's, but if the Broadband part of the circuit is enhanced (ie- a lot quicker speeds and stability) by the removal of star-wiring, then you have a happy end-user without cause for further complaint, and one who will likely be happy to use word-of-mouth to let people know what a great service such-and-such SP provide. Would the SP pass on the £160 cost in this situation ? My guess is they would stand the cost themselves.
The SFI engineer would provide an NTE (demarcation point) as part of the job.