Some ancient authors say, that Caracalla did it, because he liked to tax the provincials with the roman inheritance tax. And some modern historians trend to agree.
Others say, that it made not really a big difference, because roman citizen rights were widely spread anyways at this point of time.
And we don't know exactly about the origin of the illyrian emperors and others. This part of roman history is still heavily disputed amongst historians. Perhaps we are again, just a victim of roman propaganda. And furthermore, it was not that hard for a peregrine noble or rich family to become a roman since centuries.
The augustean reforms, which led to a stronger participation of the class of the equites in the roman society, politics and military had a much bigger impact on the dramatic change in the 3rd century. Actually Caracallas law came too late to play a role in this important process. The equites took over the empire anyways. Surprisingly, this new leading class was then destroyed in the 4th century. The luckier equites became late roman senators, the others disappeared.
I am convinced, that the Constitutio Antoniana had an impact on roman society and military. But probably it is overestimated. Looking to the late roman empires recruitment problems, there are multiple reasons bandied out. The loss of the carrot "Citizen Rights" might just be of minor importance.
PS: Some interesting read, even if not in english:
B. Pferdehirt, M. Scholz (Hrsg.),
Bürgerrecht und Krise – Die Constitutio Antoniniana 212 n. Chr. und ihre innenpolitischen Folgen. Mosaiksteine, Forschungen am RGZM 9 (Mainz 2012)
ISBN 978-3-88467-195-5