@Riot, I have a feeling you've already thought of this, but for a team to not have a marksman.....
....melees need the ability to siege.
This is the number 1 reason why marksmen, even at their weakest, are still completely mandatory.
You can have teamfights a plenty without a marksman, you can take Dragon, Baron, and Herald without marksman......but taking turrets is a whole different ball game. The range marksmen provide allows them to chip away at turrets little by little. I'm certain you already know this, so, I won't go into too much detail.
So...if there is anything I would suggest....is to make our little forgotten friend Ohmwrecker
into a melee only, turret wrecking machine. Remember those turret lasers you once had? Yeah, you can make it to where Ohmwrecker has a channel time, and once it's over, your champ fires out a fricken laser at the turret for a couple seconds.
However, this effect needs to have a VERY low cooldown in order to compete with the consistency that marksmen have. So, to make sure that Ohmwrecker doesn't become the most OP thing in the world, this laser it gives you actually drains you of whatever resource your champ uses (health resource and no resource champs will have health drained.....and at a pretty significant amount....like 8 to 10% max health total per cast).
This should allow melees to siege, and due to the channel time, it can be interrupted. So, not only do you make it reasonable to not use a marksman, but you also make Ohmwrecker viable! You kill 2 birds with 1 stone!