Speakers are Speakers?
No - everything i've read says instrument speakers are designed for musical instruments - whilst audio speakers are designed for audio systems -apparently totally different requirements and responses.
Yeah. Hi-fi speakers are supposed to do what they do and not vary. The majority of our hi-fi rigs use foam surround woofers. Foam surrounds are as compliant as they'll get when they're brand new unless we're talking about '70s Advents or similar. Old foam surrounds get brittle. Newer surrounds should age more gracefully than the old stuff.
Guitar speakers are '50s technology. They use paper cones. A lot of the cone movement is at the outer edge. Flex paper and it becomes more pliable.
All the major speaker manufacturers starting with Western-Electric and progressing through Altec, JBL and Electro-Voice (among many others) found that if they treated the edges of their cones (with "dope") the speakers were less susceptible to tearing in that area. Look at old Jensens with their plain paper cones. If they've been used hard they start to get little tears around the edges. Time for some toilet paper and glue!
Tube amps can benefit from a break-in or burn-in period. The best place for this to happen is at the manufacturer. A 12 hour burn-in at full volume in a test cell is advisable. Some of the very few manufacturers who perform a burn-in do it into a dummy load. The amp runs at full power, if anything bad happens the tech will (hopefully) see it on a 'scope. Anyone can burn their amp in by connecting it to an audio source at the correct input level. If it's done through a speaker rather than a dummy load the speaker breaks in and you can hear it if anything fails.
That's most of the point of a burn-in. The vast majority of tube failures fall into two failure modes: 1) DOA which means the tube is junk right out of the box, no doubt about those or 2) "infant mortality" which means that the tube fails within the first couple hours of use. There's a pretty good chance that tubes that survive a 12 hour full blast burn-in will survive your first year or tow of gigs.
Otherwise, pull it out of the box, take it to the gig and hope the manufacturer has a decent warrantee.