We're a HE family just outside of Newcastle under Lyme
I can't comment on racism generally, as I've not encountered it either directly or indirectly (white family), but there is a Caribbean community within the city, as well as more diverse pockets in places. It is a fairly white working class city, however. The BNP and Brexit of a few years back - and Labour councillors losing to Tories since - would potentially worry me in your position.
Of the six towns, I'd generally avoid Burslem, Stoke, Fenton and Tunstall, as they're all pretty dead and don't get much investment in them. Hanley has more going for it, but has areas of real deprivation. Longton is similar, but has more facility and a railway station - but is further out from Hanley. Near to the hospital (Hartshill) and Staffs Uni (Shelton) are more diverse generally. Likewise, near to Keele Uni, though Keele village itself isn't especially big.
Birches Head is relatively nice, in my experience. Trentham, as already mentioned (though I believe it's technically Not Stoke). I love Newcastle, though it suffers from being in the shadow of SoT a little.
For the HE aspect, there is a lovely community here, though you may be aware that Stoke on Trent and Staffordshire both have local authorities that like to overstep beyond their remit. There is currently a decent drama group that meets in Biddulph, that could be of interest to your daughter. Plenty of social groups restarting now restrictions are easing - it's not the most active HE community in the country, but there is a fair bit on regularly, and over the border in Cheshire, too.
I would probably be tempted to stay in the city for a week or two to get a feel for it, if you can. Explore different areas, and so on. Most HE groups I'm aware of would be happy to have you even if you were just there that week, which would also introduce you to that community a bit, etc.