Although I'm not a Starmer fan, the actual reason he doesn't want to publicly re-open the topic is that he's smart enough to realize that it would be a disaster for the pro-EU left.

Right now the Brexit debate in the UK is frozen in time. Remainers in particular haven't changed their ideas about it since 2016. If the topic were to be reopened in a big way (e.g. second referendum) then people would become aware of or reminded of many things the establishment would prefer people forget.

For example, this claim by Khan is economic. Remainers generally aren't aware of how badly their economic predictions failed, so continue to believe this kind of claim. But if Starmer were to make a big public position out of trying to undo Brexit, the people who campaigned for it would reawaken and refocus on it (e.g. Farage). They are more popular than Starmer and people listen to them. They would then systematically revisit the claims of the pro-EU establishment and point out how drastically wrong they were about everything. They would imply or outright assert that it was deliberate. And a lot of people - the majority - would agree with them, given what's happened in the years since.

For example, Remainer MPs promised the public there would be a massive recession at the moment of leaving caused by uncertainity. The economy grew instead. You can't be more wrong than that. Then they promised that actually leaving would be chaotic. Most people didn't even notice anything had changed. Then they promised that it would be destructive to trade with the EU, yet trade ratios haven't changed [1]. Then they promised that the UK would be frozen out of international programmes, be "alone" without friends, and sneered at Brexiteer claims that cooperation would continue as being "cakeism". In fact the UK has rejoined the few European programmes it actually wanted, like the Horizon research programme, something that academics and the EU formerly assured Brits would be impossible.

Even the guy who negotiated Brexit for the EU side has adopted Brexiteer positions in the years since.

https://www.theguardian.com/world/2021/sep/26/michel-barnier...

But the politics of Michel Barnier have, it is fair to say, moved on. Tough controls on immigration; a restricted role for European courts and a new politics of patriotism: these are the eyebrow-raising new demands of the EU’s former chief Brexit negotiator.

[...]

“The first chapter of my book is entitled ‘A warning’,” says Barnier, holding up an elegant yellow-jacketed French edition. “People in the bubble of Brussels think they are always right,” he says. “They don’t want to listen. They don’t want to change anything. This is precisely the way to provoke more Brexits elsewhere in Europe. I’m not a federalist. I’ve never been a federalist. I’m a Gaullist and I’m still on the same track – a patriot and a European. And I’m a European in addition to being a patriot and not instead of being a patriot. As the former Brexit negotiator and as a French politician I will draw the lessons of Brexit, OK?”

Remainers are for now happy stewing in their 2016-era bubble where none of this ever happened. Starmer would like to keep them on side, which means keeping them there, which means being very careful about what he does or does not say in public.

[1] https://researchbriefings.files.parliament.uk/documents/CBP-... (p15)