I'm listening to the really excellent The History of Rome podcast by Mike Duncan. When he discussed the endorsement and eventual adoption of Christianity by Constantine, he outlines a number of factors that make Constantine's endorsement of Christianity make sense. For instance, it was politically expedent to align himself with an already powerful minority during the civil wars he fought against anti-Christian opponents, Christianity was a unifying element for a political regime that had been tending towards unification for centuries, and of course he may have had the good old superstitious desire to be on the right side of the right god.
What does not make any sense at all is why in the world he allowed the Church to maintain and entrench a heirarchy of power that was parallel to and seperate from the power of the state. The old Roman state religion, if not literally an arm of the government, overlapped neatly with government power, with the Pontifex Maximus being a member of some prominent family, and the cult of the deified emperor being the next best thing to literally worshipping the sitting emperor as a god. There was no tension between church and state because wherever these were not actually the same people- as in Julius Caesar- they were people from the same families and both of these overlapping hierarchies promoted the ideas that supporting the state was supporting the gods and that the favor of the gods could be seen in the strength of the state.
With Christianity however, this will change, and in a few hundred years, the divergence between the state and religious heirarchies of power will be so great that they can actually make war upon each other. Of course, this wasn't Constantine's intent, but how could he not have seen the seeds of the eventual Pope vs Kings conflicts that will characterize later European history? When he endorsed Christianity, why did he not also wholly coopt it and, for instance, give himself the right to appoint all bishops and to interperate scripture in the manner of the Sybeline Oracles? Every other action of his seems characterized by rutheless practicality intended to cement his own power. Why allow it to wander away?
edit: Thank you for your responses.
I think there are good reasons for Constantine to act the way he did and not the way you are proposing. First of all, the Christian church of the early 4th century was very different from that of medieval times or even the 5th and 6th century. Its bishops hadn't nearly the same social standing as they would later have. They almost always were from families of medium wealth that would have been looked down upon by the old elites of the empire which viewed themselves as "the better part of the human race". Their position as high ranking christian clerics didn't confer much authority yet especially as much of the population of the empire still wasn't christian. The church as a whole wasn't nearly as wealthy as it would later become through pious donations and was still very much dependend on imperial generosity. It also wasn't as united as the medieval Latin church would later be under the leadership of the Pope. So there was no real reason for Constantine to view the christian clergy as a threat to his own authority.
On the other hand the conflicts between Pope and Emperor that you address did happen at least seven centuries in Constantine's future and depended very much on conditions that were in no way foreseeable for Constantine, like the fall of the Western Roman Empire, the rise in papal authority and it's role in the reestablishment of the imperial office in the west or the reform movement from Cluny. He couldn't prevent those events because he had simply no way of knowing about them. In fact if Constantine would have made any predictions at all they probably would have looked much more like the conditions in the Byzantine Empire where the office of the emperor had continiued to exist without interruption. And it's very telling that in the east imperial and episcopal authority never clashed in the same way or intensity as the did in the west. Here the emperor had a much higher level of controle over ecclesiastical affairs.
Last but not least the course of action you propose for Constantine probably would't have worked out as well as you might think. The micromanaging of every episcopal appointment in every city of the entire empire would have been quite a huge task for the imperial administration. In contrast the empire never had this level of control over its pagan priesthood. The emperor never personally chose who would become high priest of Elagabal in Emesa or of the imperial cult in Ephesos. Those positions were filled by the consensus of the local elites just like the appointment of bishops would normally be settled on a local level. To infringe to much on the rights of the local congregations might even have driven large parts of the christian church into opposition to Constantine's rule. By this point Christianity was very much used to resisting governmental demands that it perceived as unjust. The splitting off of the Donatist Church of North Africa after Constantine had supported their opponents is just one example of what could have easily happened elsewhere.
To claim the sole right to interpret the holy scripture probably wouldn't have worked either. Constantine would have needed a much higher level of familiarity not only with the different books of the bible but also the works of the church fathers than he actually displayed. The fact that at the beginning of the Arrian controversy he suggested to both parties to basically stop arguing over what he viewed as semantics goes to show that he wasn't much of a theologian. And even if Constantine would have been able to form his own interpretation of the holy scriptures there wouldn't have been a way to force it on everybody else. Scripture can be studied by a wide range of people and if Constantine's interpretation would have deviated to much from the consensus of the majority it would just have been vied as heresy. That's why Constantine convened meetings like the council of Nicaea. He wanted to find out what the consensus was among the bishops on certain religious matters. Once he knew he could try to put through the decisions that were reached by the bishops. This is basically how every other late antique emperor would act in those matters. It was much easier for them to find out what position was favoured by the majority of the bishops and then adopt it for themselves than to just impose their own will. This way they could easily legitimize their rule as guardians of the true faith which went a long way to cement their power.