Your instincts are correct....
The reason i asked for model number was to look it up and to see if there were other elements.
is this the top open element used as a grill function too ? careful if so as i believe there is a horizontal roller fan built into these ovens to circulate air from below, up and over the element, these can seize, fail and cause the elements to overheat.
Hotpoint cover is 5 years or it was, free parts but you pay labour - i know in 2017 it was £90 so now probably around the £120 mark.
get it replaced anyway, do a continuity check after best you can for earth in particular, ensure spade connectors that attach to elements are tight too.
When it's all done report back and invite everyone over for a roast dinner ?