As mentioned in this answer, one can set fallback cursors, if the browser doesn't support e. g. cursor:grab;. My problem now is to get these images. A local search on my drive for ".cur" just gave me the system cursors, where grab.cur and grabbing.cur are not present. A web search also wasn't successfull.
How can I get the browser cursor images?
Given cursors are usually provided by the OS (I think?) there are likely a few different variations for each operating system.
Your best bet is finding a re-created icon set for mac OS, Windows or Linux that includes the open and closed hand icons and creating your own .cur file, of which there are plenty of online conversion tools for GIF or PNG to CUR.
I found this repo on Github that provides the mac OS ones (or a pretty decent reproduction of them).
Remember that if your CSS cursor fallback ends up being used, it's likely the user has not encountered it often and they probably wont compare it to anything else in their head. The problem is your fallback cursor icon should ideally match the icon style as their system so they're not like "Huh that icon is weird."