I would find a Birmingham New Street to Moor Street dedicated connection very useful as the alternative rail option of changing at Smethwick Galton Bridge or tram to Jewellery Quarter both add significant time and additional cost to the journey. But I'm not sure its realistic to expect such a link to be built.
The St Martin's Queensway tunnel is the shortest route, largely straight and level, but generally awful: narrow, poorly lit, neglected, frequented by beggars, drug users, etc, so likely rarely patrolled.
Far be it from me to suggest that owner Hammerson likes to keep it that way to encourage people changing trains to get lost in their shopping centres above instead!
I'd like to see traffic removed, except electric buses, with a signalled single lane remaining, leaving space for a significantly widened pedestrian path alongside, with a segregating wall throughout, and ideally equipped with a moving walkway for at least part of the length. If they made it wide enough down there, Hammerson might even squeeze in some high-revenue coffee and travel goods shops to extract a levy on interchanging passengers at that level instead of trying to force them upstairs, which many can't manage anyway.
A 'One Station' project linking Birmingham's main line terminals needs to focus clearly on making that interchange as quick and pleasant as possible, because that's what the vast majority of passengers making regular routine journeys want, and is the fundamental measure of an effective interchange. By contrast, shopping centre managers live in an alternative reality where everyone has hours to kill transfering across major cities, taking lunch, buying some clothes, having some experiences en route before embarking on the transcontinental express to the grand tour. Yes life's a journey, but the commute's a grind, and the last thing anyone wants is to be diverted an extra few hundred metres round past a lot of gaudy storefronts when you just want to get your train home and definitely haven't any spare cash in your pocket.