Partly, but again that is a broad interpretation of democracy and legitimacy that in the UK context doesn't quite fit to have any significant practical meaning (again morally- i'm with you, you know how much i favour a greater democratization of Britain). Indeed Historically Westminster doesn't have a very good innings with 'trust' generally, beyond at various points in the immediate post-war era (The social consensus while not actually being a 'thing' was an era where the electorate were happy for the government to take on greater responsibility). But more particularly i would argue it relies on the electorates lethargy/not caring (hence why fewer people voting has never been a problem for the Westminster system, note that campaigns to get people to break their voting virginity, are always from parties who need the extra votes usually), trust isn't the key ingredient for the UK's 'democratic legitimacy'. There is indeed a valid argument that British politics is based not on voting for the party you support, but voting on the party you least dislike.
For the Westminster system to work and be viable, it need only make sure certain key parts of the electorate are 'meh' with it, or not in outright conflict with it (See the divide and conquer tactics used against the Chartists, or indeed the Unions much later). If a segment of the electorate feels betrayed, it matters little to it. This is the issue with brexit- most leavers quite fairly state if the result does not go through, will never vote again. Very few people will take to the streets comparatively, fewer still will cause violence (and if violence happens- that is where Britain's managed democracy thrives alas). What's more though again the factionated nature of brexit comes into play, some leavers might be ok with a customs union (Que my parents
), other's might find it not a brexit at all, the issue being they are too factionated to make this into any kind of 'threat' to the Westminster's legitimacy in any real meaningful way. The same goes for remain too. We're essentially talking about two very vocal churches, consisting of a myriad of small factions in disagreement with each other and internally, but who overall are both a minority of the electorate and thus are not enough to rock it or proclaim that Westminster and its political structure has 'no trust'.
To put it more specifically you (again morally fairly) note-
Offers to who though? Which key faction of reaminers or brexiteers is it unfit for purpose for, and moreover why does it being unfit for that particular factions interpretation of brexit mean that the Westminster system has lost trust? Whose trust? What will they do about it? Leavers and reamainers have various interpretations of their own position and each others, for your statement to have any impact of the Westminster system, this has to hold true to all of them, and that is something though that the Westminister system is actively designed to prevent, a people's voice at a GE about it is diluted and their further factionated on other issues- so whose 'unfit for purpose?'- its not, because it was never designed to be for any particular electoral bodies purpose, the whole thing is meant to be so that it can and does act in whatever way it see's fit, and is protected from any fallout from this (unless its incredibly poorly managed- but not even May is at that level yet as their has been 0 clamour for Parliamentary reform, Parliament indeed has co-opted brexit into being about 'Parliamentary sovereignty' ironically, which secures it- as people who feel that was key to brexit are essentially voting/fighting for Parliament to be able to do this to them more effectively, especially against the increasing power of the British courts which were potentially in danger of undermining parliaments sole power and giving the British people some kind of constitution that is more permanent than current.
So morally- i agree with your premise, but in practice this isn't being insipidly legalistic, this is how it actually works. What are we going to do about it?