Are scripts like sp_BlitzIndex and the others accurate on restored backups?
BlitzIndex will return some useful information on a non-production server as it is analyzing various things, some of which have nothing to do with what queries are running.  It should be clear to you when viewing the output what is relevant and what is not.

BlitzCache will not return useful information (or not much, at best) on a non-production server unless you run the same queries and load that are on the production server.

To answer your question as to "why not?", run them, review the output, and read the info at the URL returned in the URL column.  There is way too much info returned by these tools to give you a line-by-line description of why it would or wouldn't be relevant on a non-production server.