I'm don't have any real problem with it as a hobby that people like to engage in. It's certainly interesting and fun to watch. I do find problems when people insist that what they've done in sparring training has some kind of historical validity. Most of the books these groups pull their evidence from are fifteenth century or later dueling guides which is a pretty narrow field to try and then apply it to medieval combat at large.

This is not to say that I'm opposed to experimental archaeology, and this could be lumped in with that field to some extent. I've definitely even considered doing a bit of my own if I had the money and I've read my fair share of other people's works. It's just that EA is a tricky field and drawing conclusions from experiments is not exactly straightforward. This may be a bias put in place by my personal encounters with local SCA people, along with local LARPers as the groups overlap a lot here, but I have found a sad lack of engagement with the problems of their methods. Good things can be taken from attempts to recreate the past but it's a tricky business and in my experience at least the revival movements haven't been great at knowing their own limits.

I think it's great, and members of the community have contributed a great deal to the field of European military history in the past couple of decades. /u/Valkine mentioned a few problems that I agree with, although I feel that his attitude in general towards the movement is too negative.

As he stated, the treatises are centered around the 14th, 15th, and 16th centuries so any recreation of combat from the 13th century or earlier is mostly speculation based on these later works. It's understandable that someone who studies the Crusades would be frustrated at the application of 15th or 16th century texts to the martial arts employed in the First Crusade, which may have been similar or may have been significantly different.

Often hobbyists will conflate the various martial arts of different periods, regions, and masters into one pan-European martial art. It's easy to talk about "historical European martial arts" when differentiating them from Asian martial arts, but there was a huge amount of variance within Europe's martial arts across periods and regions. While significant similarities between the treatises are apparent, and there are identifiable martial traditions, there are also significant differences and distinct styles. Fiore's Italian longsword tradition is different from Liechtenauer's German longsword tradition and a combination of these styles, while perfectly reasonable and effective for a hobbyist's sparring purposes, should be treated as an ahistorical anachronism. The authors of the various treatises often had different, and sometimes contradictory, opinions and so each treatise should be treated as a unique style and practiced as unique styles if any pretense of historical accuracy is to be kept. The treatises differ wildly on even seemingly simple matters such as how to grip a sword.

The historical setting of the treatises must also be taken into consideration. Liechtenauer's style of longsword was originally a special style, with secret cuts and techniques to defeat a typical longsword fencer. His techniques were later popularised by masters such as Ringeck and Danzig, and the previously "secret" cuts became the "master" cuts, but many modern practitioners make the mistake of assuming that Liechtenauer's style was always the typical or only style.

HEMA as a movement definitely should not be dismissed just because some SCA or LARPer hobbyists care more about enjoying themselves than historical research. We have great people who I would consider to be "true" HEMA historians such as Dr. Tobias Capwell, Mike Loades, and Matt Easton doing their best to study the evidence in its historical setting and recreate the martial arts as authentically as possible, in the process contributing to the study of European history as a whole. Just to be clear, I have absolutely no problem with SCA, BoN, LARPers, etc. Not everyone is interested in rigorous historical research, nor should they be expected to be. However, as Valkine pointed out, it's frustrating when someone thinks they are a historical martial arts expert just because they have some experience sparring with padded rattan weapons. Even if a person were to practice enough with historical weaponry to be become a modern master, without supporting historical evidence they could only be regarded as having developed a modern style using historical weapons, and not having rediscovered a true historical martial art. A lot of modern practitioners (of martial arts in general, not just historical ones) make the mistake of believing there's only a single true art that all martial artists would eventually converge on given enough experience, which is certainly not the case.