As you have found out, as I did, when you actually contacted these 'immediate' house purchasing companies, they make such a large discount, that it is impossible.
I put my house up, through EA's but insisted it was marketed at the very bottom of the sort of price for that sort of property as I wanted a reasonably quick sale. Then, a few weeks later I actually reduced it further. When I telephoned one of these companies, all they do is to check on something like RightMove on the price is it being marketed and then take off between 10 - 20 percent of that price and that is what they offer you. At the time I did say to them, that if I was intending to use them, I would prepare by actually marketing my problem above its value, just so that they could then base their price on that.
Unless someone is absolutely desperate this is a dreadful idea.
The stress caused by your situation is, sadly, so normal with the very bad system in England and Wales. It has even been known in the past when the problem is with someone at the bottom of a long chain for the sellers at the top of that chain actually purchasing that other property themselves, just to allow the chain to proceed. Obviously not often a possibility.
All I can say, is stick in there. It will all happen -and when it does it will be virtually instantaneous and all systems 'Go'. To pull out, just to re-start is not a good idea. Seven months along you could still be in a similar position. All the solicitors in the chain should be really chasing this, every day - and putting pressure on that final mortgage offerers.
I hear your story so often - and in nine out of ten cases, it does all finally come to a happy end. Sure you will be one of those. GOOD LUCK