OK, I've thought about it.
So we're probably dealing with the Old Republic for the creation of the Adeptus Jedi. (tm)
-means we have the Rights of Sentients (outlawed slavery)
-pretty much means the Hutts are gonna get boned along with any other Slaving-running species
-Wars with the Mandaloreans are probably going to be shorter and bloodier.
-Sith will be seen as Terrorist organization (drone strikes a plenty) or just hit teams of Adeptus Jedi
Effects of these very militant protectors on follow on forces:
-imagine that ANY Adeptus Jedi would have the authority to command whatever military/naval forces were at hand to deal with whatever that Adeptus Jedi needed dealt with.
-ability to deputize others as necessary for the pursuit of fugitives and criminals
-possibly even the dispensation of Justice in all sectors; and I use the capital (J)ustice meaningfully. Their decision would have the binding force of law though I'd imagine there would be some path for appeal to a specific authority. (maybe a sector political officer or court)
Just running through the numbers:
Old Republic was formed roughly 25,000 BBY for a starting Adeptus Jedi population of 10,000 + "Staff"
-it's a rough job and there's lots of galaxy for accidents to happen so let's just say that they loose 1 a month. That means they lose 120 Adeptus Jedi every 10 years on average then DOUBLE that just cuz. So -240 every 10 years. (I'm being kinda pessimistic hear) We'll call a Company equal to 100, and say there are 4 Companies to a Battalion. With that in mind that's 400 new Adeptus Custodes every 10 year - 240 lost for the same time will give a growth rate of 160 every 10 years.
So...by the time of Senator Palpatine they MAY be upwards of (160)(2,500)+10,000 = 410,000 Adeptus Jedi. Now...that's just kinda nuts to think about, and disturbing for every member of the Senate and Galaxy to realize there are people as old as the Republic about! (Though we do know that there are Adeptus Custodes older than 10,000 years, so it's not so crazy in fluff) To make it work a bit better, I can see the Adeptus Jedi have their "Senior Citizens" retire to hidden location/s to continue to train and support the 'active' personnel. Some are sent to hidden fleet bases, or where the gene-seed is kept.