No.
While it's been rightly said that the mini is not appreciably faster than a MacBook based on broadly the same architecture, it's a bizarre conclusion that that makes it unsuitable for video editing work. It's not the
best system for video to be sure, but users have been running various versions of iMovie, Final Cut Express and Pro and Avid and Adobe video apps on Macs that are much slower than the current minis, and had little in the way of complaint while doing so.
As such, while for this user, if a MacBook is insufficient for the task a Mac mini is hardly likely to be better (excepting that it seems likely the MacBook has issues preventing it from delivering best performance) in more general terms the current Mac mini is perfectly capable of this kind of use. It may be somewhat out of date in architectural terms, but then by and large the software is not exactly cutting edge (or particularly demanding) either. And while the mini is the bottom-end system and thus the least capable of any current Mac, it has performance that is broadly the same (actually marginally better) than a first generation G5 dual processor powermac, many of which are still in daily use in professional video production environments.
For this particular user, there seems little doubt that examining the reasons the MacBook seems insufficient is a better place to start than spending money on replacing it, but in the wider sense it would be wrong to give the impression that the Mac mini is not suited to this sort of use.