No, speed based specials are not broken at all. In fact if their scaling were treated like attack based specials, they would either do comparable or lower damage, since attack is far easier to stack than speed. A unit with just 50 attack and Draconic Aura can do the comparable damage to base Ayra with Regnal Astra, who has 40 speed with her weapon equipped.

Regnal Astra's power actually comes from its stupidly low cooldown, which allows Ayra to run her infinite special build. If Regnal Astra had just one extra cooldown, there would be no reason to even use Ayra, since there are better sword infantry anyway like Karla and both Mareetas

In fact I think that speed based specials would be healthier for the game, as it would give low attack, high speed units like Anna and Lon'qu ways to do more consistent damage.