Therefore I would argue that if immigration is what concerns people most, then our leaving the EU must also be a part of that conversation.
Maybe if Remain campaigns had been willing to address that before the vote in any meaningful way, sure, but they didn't, part of their failure was thinking 'immigration is good, look at these lovely successful migrants, anyone who thinks otherwise is a bigot' was going to do anything other than help the Leave campaign. At this point, that assertion isn't much - sure it can be part of it, but that's too little, too late.
Any discussion of immigration needs to focus on the changes to the systems that have happened over the last few decades around non-EU migration, why those changes happened, and why they weren't solutions for British communities. Non-EU migration has always been significantly larger, that's part of why we were used to test out biometric IDs, digital IDs, and the private partners now being used in age verification and the push towards expanding digitising government services including digital IDs. All of that started in migration processes for non-EU migrants.
There were hundreds of changes to immigration law between 1997 and the referendum, the changes kept happening afterwards, no one seems happy with the outcomes of those changes other than those making money off of it - and that's from the private partners behind the software used to process immigration paperwork & maintaining digital IDs to the gangs that bring people in illegally and make quite a bit off both forcing them into labour to pay off debts and bringing them back in when they're deported. In my work, I've met people who've been through this several times, and all the ones I've met have been from the EU. The gangs pre-date Brexit, the private partners making money off of immigration pre-date Brexit, Remain politicians could have dug into these and many other immigration issues and openly discussed solutions that could be done while in the EU - but quite a few of those politicians were the ones who brought in those private partners, they discussed in Parliament that they were making the immigration services 'for profit', and seemed more than willing to leave discussion on EU gangs with the Leave campaign rather than discuss potential solutions. They would have to admit that their previous solutions were wrong, they would have to admit they were wrong to blame the EU for so many things that were actually in their control was incorrect, and they chose not to do so. That was a failure on their part, an active choice they made, that led to how things are now.
Returning to the EU would be a massive undertaking, and until those in power who are for it are willing to recognise those failures and the part they played in them - rather than just calling people stupid - none are going to be in the position to handle it. They need to get a handle on that and a lot more shit going on here before they move towards rejoining the EU.