Setting aside that your strategy goes against narrative logic and would apparently allow a high level fighter to draw and stow 8-10 times in one turn if they action surge, which is ludicrous, it also clearly goes against the intent of what is happening with cleave.
I think they can draw or stow 8-10 weapons in one turn. Every time they make an attack with it as part of their attack action they can draw or stow one in my understanding of the new rule. Cleave is an attack.
There are many threads online talking about how you can use dual wielding with nick and get 4 attacks in a turn while still wielding a shield. You need to switch weapons multiple times to make that happen.
So let's go to first level and show some examples.
A 1st level fighter using a single greatsword can't Draw-Attack-Stow because he is only making one attack and can only do one of those. So if he is empty handed at the start of his turn it is:
Draw greatsword -attack and he ends the attack action with sword in hand (and RAW can't even drop it). IF he started with it in hand he could stow it after the attack and end empty handed.
If same fighter is wielding a greataxe AND there is no other creature within 5 feet of the first and with his reach it is the same:
Draw Greatsword-Attack end the turn with it in hand
However if there is another creature within reach and within 5 feet it is now two attacks so he could
Draw-Attack-Attack(Cleave)-stow
At 5th level this fighter gets extra attack which puts another draw/stow in there so he could:
Draw-Attack ...... Attack(Cleave)-stow ..... draw-extra attack dagger .... draw-nick attack scimitar. If this character has Dual Wielding he could also get another attack with either the scimitar or the dagger as a bonus action.
Note besed on what I read, I do not believe you can draw or stow when you make an attack as part of somethign other than your action. So on bonus action attacks and reactions the weapon must already be in hand.
I am not sure if you get one free interaction in addition to all these draws and stows that come with attacks.
I don't have a copy of the PHB but that is my understanding.
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