There are a good amount of action games that do benefit a lot by only having one or two melee weapons. Granted, with action games like Killer Is Dead, No More Heroes III, and/or Oneechanbara Z2: Chaos, each game does something unique or different to compensate. Though usually, your character or characters better have a good sized or large move set that is useful and effective. Having good to great level or even straightforward level design helps too of course.
Killer Is Dead's main character, Mondo, he only has a katana and a cybernetic right arm called Muscle Back used for melee. Both his melee weapons can be upgraded to be more powerful in damage and to gain new attacks. In addition, his cyber arm can transform into a gun and he has three other forms of the gun he can unlock. Though really, only the first gun and the laser gun are most viable and practical. The ice is pointless and the drill you will use to breakdown barricades for hidden items/collectibles. Once you upgrade the charge punch, the drill becomes useless in combat. Also, Mondo has counter where if you dodge at the last minute, a gun wielding enemies bullets, you can perform a counter shot that will instantly kill them. It is an upgrade you do have to buy. Good news is that you can go to the upgrade menu anytime.
Travis does have a nice size move set in NMHIII, but it is stripped down compared to Desperate Struggle. His beam katana can now morph based on your button inputs, though he can no long dual wield Rose Nasty. He is move upgrades are overly expensive and there is not many compared to what came before. Something I do disagree with. To make up for it, he has the Death Glove that allows 4 different special attacks (you only start with 1 and get more as you progress through the game) that work on a cool down. You can upgrade chips the make the cool down be shorter. He's get fuck everything screen clear, if he let's on the special bar roulette, and later unlocks a super form that has a similar screen clear, if the player decides to use it.
Oneechanbara Z2 allows you to switch between 4 characters (Kagura, Saaya, Aya, and Saki) on the fly, and lets you do it mid-combo if you're good enough to time the attack strings right. There is a rythym to it, similar to the Cool Combos system and you can ever carry over the hard to do Cool Combos when switching characters. Each character has two melee weapons and one sub-weapon. The only exeception is Saya, who has her own dedicated charged parry button where the sub-weapon would be. So it's like having 8 weapons total when combing all the characters together. Each lady gets their own berserk mode and later super mode too. Each one also same magic blue meter with three different styles of attacks. Aya and Saki have a slowdown time dodge only they can do, while Kagura and Saaya can heal faster because they're dhampir.
Oneechanbara Origin is similar to Z2, execpt you can only switch between two characters again, and move set is smaller compared the last game. In place, universal parrying exists by having a dedicated parry button. It's really useful and powerful.