This is rather easy to address, really.
The solution? Don't remove flash, modify it: For X seconds after using a dash-type ability, Flash is disabled (let's say 5 seconds to not overdo it, but it should probably be a bit more in reality, something like 7-8). For the same X seconds after using Flash, dash-type abilities are disabled (similar to the inability to use use flash/dash-type abilities while snared, only without the whole 'you can't even walk' aspect of snares).
Champions without their own dashes see absolutely no downside, champions with their own dashes can no longer use them in any remotely chained fashion with Flash, causing it to be more situational and last-resort-oriented to them, and making other spells a little better by comparison.
Hell, it could even get other aspects buffed (maybe a small drop to the cd and a bit of its range back) if that were to happen.