A Legion society through this change in doctrine would certainly shape the conventions of the organisation, with an emphasis upon cultural unity by cooperation and diplomacy. As with Caesar's alignment with the Followers of the Apocalypse, the dispense of an education institution, higher protection on the area (though this is done effectively in the current timeline), and a general emphasis upon ingenuity would re-shape the entire composition of society. As such, the primitive tribes would grasp an overall wider scope in knowledge, whereas the current timeline actively abolished the population in gaining an education outside of the Legion dogma, especially from the external world.
Within a society more tolerate to discovery and knowledge, the Legion could possibly be a force to develop infrastructure, equipment (high quality) and technology. Combined with how the aforementioned would imbue security and a level of improvement to the Wasteland, the projected image of the Legion would be seen in a more positive light. The barbaric strategy would be dismantled for cooperation, or at least a de-emphasis upon the sadistic method. Because of the change in doctrine of the Legion society, it can lead towards the configuration being a large conglomerate of local tribes, under the perceived wise and magnanimous Caesar.
Consequently, though, engenders the notion of sustainability, an aspect of the Legion that is heavily scrutinised (for good reason). If the Legion was, as I've stated, a conglomerate of semi-independent tribes, it could possibly open the opportunity to attain more political members in administrative body. Whilst it wouldn't be a democracy - far from it - it suggests that there could be more political manoeuvring, with members all vying to assert influence to their tribe. Assuming that the megalomaniac personality of Caesar would allow that to occur, it would witness divisions and alternative perspectives. As for the leadership itself, I imagine that it would persist of the prevailing hereditary system; it is thus capacious and dependent upon the leader - though the full authority would have to factor the increasingly educated and, with time, politically savvy population.
I imagine that the campaign of the Mojave would be a salient priority of the Legion: the Dam and Vegas can provide the foundation to further expand the technological advancements of the nation, as well as use the Dam for purified water and electricity. In addition, it could be argued that a Legion technologically advanced and less prone to vindictiveness (at least in comparison to the current timeline) could seize the Dam, albeit if their hubris wouldn't have led to positioning the army to the devastating blow at Boulder City. If they did lose, then I believe it would have dealt overall more damage to the Legion; an educated population would understand the wider scope of the situation, and whilst that could lead to a society that adapts and changes its direction to a different goal, Caesar's determination will continue the campaign. As the gorilla tactics would either be less implemented or completely removed, I believe that the Legion would be halted in the Fort, unable to penetrate the solidified fortification of the NCR. As a result, it would either prompt another battle, which would yet again lead to defeat, or the Legion completely changes its goals, perhaps expanding East.
In summery, a benevolent Legion would magnify upon less brutal and unscrupulous tactics. The beliefs of the Followers would certainly interject an importance upon education and technology - aspects that were obliterated in turn for unequivocal fanaticism to the Caesar - and would mould a improved society. With this, not only are the primitive population will transverse into an educated society, but the Legion would be overall perceived in a less negative tone - behaving as an force of rebuilding society through the ethos of the Roman, as opposed to merely exploiting the framework in the pursuit of conquest.