I think the idea of taking the public with you can be totally legitimate in some areas.
I'm thinking of things like persuading the public to vote to legalise abortion or gay marriage in Ireland, for example.
Or Labour could have said they needed to take the public with them on the Brexit debate by putting forward a positive case for remaining in the EU. Or the SNP could say they need to take the public with them in respect of Scottish independence.
But I think there's a key difference here.
Firstly, to take abortion and gay marriage as an example, you're saying, "This is something that might not affect you but it is really important to others. It might be against your own moral values but you don't have to do it yourself, this is about freedom for other people to live their lives the way they want to as long as they aren't hurting anyone." And it's really difficult to argue that someone else having an abortion or marrying their same sex partner is hurting you personally. Or in terms of Brexit or Scottish independence, you make a powerful case for why you believe your way is the right way and let people think about it.
And secondly, these are things that were all going to be put to a vote. So yes, absolutely throw everything at "bringing the public with you" and convincing them that their fears are unfounded and this is the right side of history, if you are going to allow them to vote on it once you've had a decent go at bringing them with you.
If Starmer said we'd have a proper debate about self ID and then a referendum on it, then maybe he could talk about bringing the public with him. But if the general public isn't getting an opportunity to vote on this - and a general election doesn't count - then you're not bringing anyone with you, you're imposing your ideology on them and not listening when they tell you how it affects them negatively.