That would be an even sillier implementation of DRS than we have now, because surely that would be equivalent to them both not having DRS at all!
I'd agree that would be better - but then it wouldn't be DRS anymore.
I think DRS is more a philosophy than a specific system, if that makes any sense - that philosophy being "a car running behind should be given an artificial advantage over the car in front", and the idea of a moving wing simply being the chosen method of achieving that. A push to pass system doesn't do this because there's equality between drivers - they all have the same amount of it, and can all use it wherevever they want. There's a different purpose behind such a system.
So suggesting a push-to-pass style implementation, whilst a great idea, isn't really a "modification" to DRS - it changes it into something else entirely.
I'm not sure if this would be an improvement, but I've always wondered what would happen if DRS worked in a sort-of opposite manner to how it does now - where the driver running behind can use it
wherever they want on the track
except for the usual DRS zones on the big straights. In my head I'd like to think this would mean they could run closer than usual to the car in front all the way round the lap (possibly allowing overtakes in more adventurous places), thus making an overtake on the long straights easier, but with the overtake itself being all their own work. Would also bring back the challenge of using DRS as early as possible out of corners like we used to see in 2011 qualifying. But with DRS being ineffective at lower speeds I don't know if this actually would work better in reality.