Why did the princeps/emperors of various dynasties prior to Constantine the Great (d.337 AD) allow the Praetorian Guard to continue to amass such power and repeatedly cause trouble, including palace coups?
It would appear that the Praetorian Guard gained their reputation for having some power over the principate, rather than the principate or dominate having power over them, with the rise of Sejanus who consolidated the Praetorian Guard into one camp under a prefecture. Prefects of the Praetorian Guard would prove to be troublesome from that time, but the Praetorian Guard remained problematic as an institution even if prefects were executed.
There were notably different units of guards who were chosen by the emperor himself to protect him. (The Imperial German Bodyguard, the Speculatores, and the and Equites Singulares). Despite this, the Praetorian Guard remained powerful.
If the Praetorian Guard and its prefecture created repeated problems then why were they not restructured or reduced? Why not circulate elite veterans of various legions or rotate the Guard?
The only compelling reason I can think of is the need to have absolute discretion and trust, but this is probably why the Princeps rotated their personal bodyguards. The Praetorian Guard were the only ones allowed to be armed in certain areas of Rome, and their loyalty was to the 'Emperor' first before the Roman Senate or People. Perhaps it was problematic to rotate other units of the legions through the Guard as the Praetorian Guard was specifically loyal to the institution of the principate.
Constantine the Great, who seemed to be adept at consolidating imperial power, finally dissolved the Praetorian Guard in around 313 AD.