I think that this idea that because they talk about a fairer society, they are representing those voters, is where people get a little mixed up.
It's all well and good to say we need a fairer society, but that isn't actually the same as representing the voices of those individuals, or listening to the specific concerns they have. It's very possible to look at a group and say, we will look out for you, but it's actually about doing the things you think they need, rather than what they think they need.
The biggest evidences of this are all of the things that have meant the LP have not been able to rely on the w/c vote in the same way. Even simple things like their discomfort with national pride, the flag, singing God Save the Queen. Mostly those aren't fraught political questions, but they represent a way in which the party is very out of step with the way the people they supposedly represent see the country. A lot of the time they seem to feel deeply embarrassed by them.
Comments about white vans, calling convern about communities changing due to immigration racism, calling people gammon, dismissing anyone who wanted to leave the EU as a racist. And actually the bizarre assumption you get from many that any concern with movement of labour is racism. There seems to be no understanding that free movement of labour has always been a wet dream of the capitalist elite, and it makes a lot of sense for w/c people to be much more skeptical about it.
You then get the LP saying the Conservatives also like movement of labour - well, yeah, but you aren't really representing the w/c position either, LP, you are calling them racists instead. Nor do they seem to understand that stable jobs and communities have been traditionally more important to workers than state supports.
What the LP mainly seem to represent is a very m/c view of what would help workers, mainly being forms of redistribution of income.
If they actually respected the w/c you wouldn't see all the gammon/racist bullshit, even if they wanted to argue for the same policy ideas.