Is The Overall Weight Of A Speaker Important?

Only in that the heavy speaker is better at geting the acoustical energy created by the drivers transferred into sounds waves instead of vibrating the cabinet or in some cases actually moving the speaker back and forth.

One of the cheapest tweeks that I currently employ is setting a very heavy weight, about 30 lbs, on top of my very compact subwoofer. It is a Velodyne that has an extremely long throw driver inside a cube barely larger than the driver itself. It turns out that the subwoofer was not heavy enough to keep it from moving back and forth when at medium or high output.

It's not like it was walking across the floor but the bass tightened up and the transient attack is noticebly better with the sub weighted down.

Don't know if cabinet flexible is a desired goal but many speaker makers do not go to great lengths to tame any resonances and just incorporate them it into the overall frequency responce of the speaker.

"they claim excess weight just stores unwanted energy"

If its unwanted energy wouldn't we want to store it someplace? If it is not stored where did this unwanted energy go?

Multiple English speaker makers have done an excellent job of using the box resonances from light enclosures to augment the bass output. The box resonance is "tuned" very much like you "tune" a port to be a positive addition to the overall frequency response.