On the show knight fight, and in many other clips I've seen from other similar leagues, weapons are rarely used the way they are shown in movies etc. Instead they will usually try to close the distance as fast as possible and use the weapon to shove/tackle their opponent to the ground. Because they are heavily armored, a strike from the weapon mainly stuns an opponent, but getting the opponent to the ground is usually what decides the fight. I get that the people who compete in these leagues are often mma fighters and such, and have limited experience using a weapon so may just prefer to rely on grappling/pummeling for that reason. But if effective weapon use could beat that it would seem someone would have figured it out and started dominating. But from all the competitions I've seen, it looks like ugly fighting (tackling/shoving/grappling) wins the day much more than actual weapon wielding. Reminds me a lot of traditional martial arts vs grappling in the early days of UFC/mma.
Did fights look closer to what we see in movies (those considered reasonably historically accurate), or more like the ugly brawls we see in armored combat leagues? If there was some kind of gentleman's agreement back in those days that kept knight combat from devolving into shoving matches you would think in actual combat someone would just disregard that and use what works best.
Also, obviously their swords are blunted etc but still. You have exactly one chance of tagging someone with say a war hammer before they have closed in on you and your weapon is useless.
Okay so there's two separate things going on here: Was grappling a large part of combat between heavily-armoured individuals in the medieval period, and why do modern imitations of such things like Knight fight seem to involve a lot of using their weapons as aids to grappling?
To answer the first question, manuals on combat from the late medieval period such as those from the german Liechtenauer tradition and the famous flower of battle by the Italian armsmaster Fiore Dei Liberi depict wrestling as being foundational to both unarmoured and armoured combat (and even give instruction on how to wrestle while on horseback).
For example, in the flower of battle (written before 1409), Fiore Dei Liberi elaborates 19 techniques to be used when fighting in armour with a sword, about 12 of which involve throwing the opponent, a joint lock, or disarming him, sometimes giving options to throw or disarm depending on how the opponent reacts.
Likewise, in his section on fighting in armour with the pollaxe (which he calls the Axe) he gives us 9 specific techniques of which 5 deal with throws or joint locks or disarms (including one which involves uses a strange weighted rope device device attached to the head of the pollaxe to wrap up the opponents legs).
Finally, 11 out of 31 plays in his section on mounted combat deal with wrestling an opponent out of his saddle or applying a joint lock or diasarm.
To give another example, 31% of the pollaxe plays described in the fighting treatise called the anonimo Bolognese (written between 1500 and 1515) end with a throw to the ground, 10% with a push or pull using the heel of the axe, 7% with a blow to the foot to unbalance, 7% with a disarm and another 7% hooking an arm, leg or the neck with the hooked spike.
So clearly grappling was considered a big part of armoured combat by the people who taught it. To give an example happening in real life, in 1386, two French knights (Jean De Carrouges and Jacques le Gris) engaged in trial by combat before the king and assembled court to resolve the accusation by Carrouges that Le Gris had raped his wife. They began on horseback, dismounted when their horses were killed/wounded, and fought with swords. Despite being stabbed through the thigh, De Carrouges threw Le Gris to the ground and, after demanding he admit to the crime, stabbed Le Gris to death when he refused to confess.
Despite this, we do have archaeological evidence that people were killed in battle despite their armour with weapons like pollaxes, such as Richard III of England. Additionally, Fiore informs us that a blow to an armoured opponent's head with a pollaxe will almost certainly be fatal (and in the case that this doesn't result in death, his advice is "hit him again")
The key part of understanding why modern Knight fight and BOTN fighters a) wrestle so much and b) fail to inflict lethal blunt-force injuries on each other is answered in your question:
But if effective weapon use could beat that it would seem someone would have figured it out and started dominating.
The rules that Knight Fight uses are taken directly from BOTN (Battle of the Nations) and these rules, for one thing, do not allow thrusting, immediately stopping competitors from using a common technique in the manuals: stab the opponent in the throat or the visor where protection is less than in other areas. The rules also disallow direct strikes to the face and to the back of the neck.
Another is the armour requirements: BOTN armour is on the heavier end or even off the end of the scale for accurate coverage and thickness of armour. A criticism by a historian of Knight Fight in particular notes that the armour worn can weigh up to 80 pounds, whereas historically even full-plate coverage weighed around 40 pounds.
Finally, BOTN rules state that it is prohibited to injure someone deliberately, this is basically all we need to know here: they are simply not going to unleash a full-strength pollaxe swing at someone's head because weapons designed to hurt someone in armour, can in fact hurt someone in armour.
The TL;DR is that late Medieval armour was very protective and wrestling was absolutely a method used to fight in armour, when the situation made lethal thrusts or blows with armour-defeating weapons impossible. As such, the rules of Knight Fight/BOTN style events prohibit use of weapons in ways that, while historically accurate, would lead to serious injury or death, leading to lots of wrestling.
Sources:
Getty Manuscript of Fior Di Battaglia Translated by Colin Hatcher avaliable at: https://wiktenauer.com/wiki/Fiore_de%27i_Liberi#Preface
Jaquet, D. 2017. The plays with the axe in armour of the Anonimo Bolognese (1510-1515). Acta Periodica Duellatorum. Vol. 5(1).
Whitbread, R.E., 2013. Tournaments, Jousts and Duels: Formal Combats in England and France, circa 1380-1440. (Doctoral dissertation, University of York).
Edited to add: I want to make clear that this isn't me ragging on BOTN or other BOHURT style medieval combat sports, I think they're pretty cool and fun to watch, and if I could afford the equipment, I might be tempted to give it a go myself. Even people who do HEMA (Historical European Martial Arts - which focuses on trying to emulate techniques depicted in the manuals) like I do, need to admit that we can not practice these martial arts in the same way people did in the past for a whole host of reasons, and particularly modern competition HEMA seems to have a never-ending debate about 'realism' and 'realistic fencing' in competitions where the rulesets and protective gear necessarily induce 'unrealistic' styles of fencing because they produce the best results from the point of view of winning a bout.
Edit part 2 (edit harder). BOTN rules also prohibits "twisting against natural direction of a limb flexion and any painful holds" as well as "any clinches, hold-downs, suffocating techniques with a weapon or hands over the neck from the front or from behind", which is why we don't see joint locks or holds that are common in manuals either.