One of Ohmwrecker's biggest problems is that it's not very useful when you are behind. This would certainly make it a lot better when behind, when you can't use it to dive enemy turrets, and also keeps its turret-focused niche.
The problem is that, when stacked, this item would basically be used to stall out sieges for a ridiculously long time. Stack it on 5 people, take turns fortifying it every minion wave, and suddenly the enemy team cannot push down your turrets because each minion wave the tower would get fortified, take no damage, kill all the minions, and then reset until the next minion wave comes. Which takes about 30 seconds(?) to come out, giving long enough time to have a near-permanent fortify on the turret; either the fortify effect is not active or there are no minions to take the turret.
What's the solution for the other team? Buy Ohmwrecker and kill the enemy team. How dare the losing team try to stall out a game indefinitely! Buy Ohmwrecker to disable the enemy turret, dive them, kill them, and then take the turret when there are no champions to fortify it.
Basically, with this kind of item in this kind of state, the game would essentially be in a state where Ohmwrecker is a must-have item on at least on person on each team, if not all five players would get it, with that kind of state.
It would stall out games, and make early game champions that fall off and champions that are weak early and late and spike in the midgame would basically be suboptimal, whereas late-game team comps would be the ideal strategy by a large margin because there is relatively low risk.