I think this idea that Corbyn doing fuck all during the Brexit campaign swung it for Leave is pretty overblown, IMO, but I really don't see Starmer as being a particularly good campaigner in that context either.
His whole thing is being boring and competent, it was a referendum launched in response to right populist pressure on the Tories and it was a very heated (to say the least - RIP Jo Cox) campaign, much more about competing interpretations and stances on ideals and ideologies (patriotism, multiculturalism, sovereignty, etc) than it was about putting out different policy positions or whatever.
And even before that, Corbyn became leader because the mainstream of Labour had become so utterly discredited after the 2015 election - there were no big ideas within the parliamentary party other than chasing the Tory and UKIP lead on reducing immigration and cracking down on benefits while failing to still actually convince anyone of the sincerity of those stances. (Why vote for the diet version of what the Tories were offering anyway?) I don't see any of the other leadership candidates from the 2015 contest doing much better than a hypothetical Starmer either as a result, other than maaaaaaybe Burnham, but that's assuming his reinvention as a bit of a left populist since becoming GM mayor reflects how he was all along, rather than another bit of post-Brexit/Corbyn triangulation from someone who was just as much a student of the New Labour heavyweights as, say, Yvette Cooper.