Much of this thread shows up the poor general state of Internet access in the UK . Most of it comes down to poor regulation allowing providers to get away with low service performance levels and leaving the customers with contracts that are hard to escape from. Basically any demonstrated poor service that lasts more than a month should allow the customer to move supplier but as always in UK business comes first.
As to the OPs problems which seem to have diminished to a degree , perhaps some HH settinngs have been altered which ahve improved things and even if they will only send you an HH5 these are not all the same - they bought a lot of them over a period of team and as with all such equipment various software/firmware fixes are incorporated in later generations.
Perhaps the formal seperation of OpenReach (chat toa BT tech if you know one about what this means for cohabiting BT employees , human rights violations all the way there) will make it easier to change but in itself thats not simple.
Where I live -in a small raod which is private but which is not a private road as we do not join onto a public highway but drive over the pavement to get to it this means we cannot get Virgin who are allowed to skip roads that dont suit them. Sky are next best as the others are too small really but I object to giving the scum who own that enterprise any money. So when it comes down to it it is BT or nothing and soem of their products are best in class. Not BT email though stay away from that unless already committed.
What you could try is write to a suitable BT exec with a brief synopsis and say you feel you have been let down by the company who put contractual detail ahead of customer service and that this will make it unlikely that you will continue with BT when your contract runs out. Make the comment that the HH5 is known to be less than perfect (a lot of the UK based BT tech support people are quite open about this if you 'chat them up' and that one way of keeping you would be to give you the HH6 ahead of your new contract because thats ikely to persuade you to stay. However a new HH5 could help so give that a try first. Good luck