The function of Jedi within the GAR were primarily as Commanders and leaders. The Jedi have a long history acting as generals, commanders and leaders on the battlefield due to their intuition, quick and often unconventional thinking, and this was a role they once again undertook during the Clone Wars.
The GAR was however bred with its own class of Clone Trooper Commanders which fulfil such roles, and a number of non-clone commanders and fleet admirals were also present within the command hierarchy of the GAR. Additionally, Jedi were utilised as front-line units and often formed the vanguard of an attack, a role for which the GAR had numerous commando units to undertake if Jedi were not present.
It should also be noted that Jedi were only ever assigned to GAR stations and campaigns across the galaxy which were considered critical, as the relatively small number of Jedi limited their deployment prospects. Most GAT positions and garrisons could be easily managed by Clone Commanders.
In short, the Jedi were a valuable asset to the GAR and boosted their overall effectiveness, however the Jedi were not critical to the functionality of the GAR, as Sheeve Palpatine (The architect of the GARs creation and the Jedi's demise) had planned for the GAR to operate in the service of the Galactic Empire after the execution of Order 66.