Prime Runner status kicks in at, what, 150-ish Karma? (It’s been a long while since I looked at those rules.) Given that you’re typically looking at 5 Karma (on the low end) to 10 Karma (on the very high end) per successful session... I’d say the average is probably six or seven. That’s approximately 30 sessions played...
Expecting a ‘runner to go that long and still see Initiation as a big deal is like asking a 12th Level D&D character to get chuffed over finding a Dagger +1.
I’d say a GM would be reasonable is limiting Awakened Prime Runners to only one or two grades of initiation, telling his Augmented Primer Runners that only 25% of their money could be spent on alphaware, and so forth. This would sort of reflect that, despite being built as experienced characters, they “really” would have be subject to a lot of limits had they been built as “regular” characters.