Whenever I read articles about probe start metal halide lighting, I I read that the lamps must be group replaced and they must not operate on their intended ballasts until they completely burn out because many of these articles say that running these lamps beyond their rated life causes damage to the ballasts. From my understanding, they will eventually damage the capacitors if they are used on CWA ballasts. Why is this so?

This is completely BS. There are no reasons to not run these lamps until they exhibit signs of failure. No manufacturer outside the US saying such a thing.

It says this in the GE lamp guides about halide lamps, but more so because of having, ( more or less ), colour uniformity, I’ve seen the results of spot replacement of 400 watt halide lamps where I use to work, and the new lamps make the old ones look awful!
It also cuts down on maintenance costs. As for damage to gear, I never witnessed any, even with exploded lamps!, as soon as the new one went in, it would fire right up.

Yeah generally the following reasons:

- colour uniformity as different brands and different ages will appear different next to each other in a large install
- lower labour costs if needed to hire someone with a lift to relamp, which in most metal halide installations is true
- significantly less risk of a non passive failure, particularly if the lamp was operated horizontally   

It is more for end user convenience. Many probe start lamps have colour shift, more noticeable towards the end of their life. In order to maintain proper colour, manufacturers recommend group replacement after a number of operating hours.

Also in large installations, group replacement means slightly more profit for the manufacturer. Lets say a large customer account relamps at 75% rated life, the manufacturer squeezes in an extra lamp change every four times, compared to another large account that waits until the complete rated life (or who spot relamps). Multiply this by a very large number of very large installations and the difference is noticeable in the manufacturer's accounting books.

As for ballast damage, that is merely an excuse to encourage earlier lamp replacement. I've seen CWA installations have EOL lamps for months (maybe years ?) and a new lamp worked just fine for years after.

It is not about group replacement per se, but about the requirement to replace the lamp before the end of its rated life. Doing it in groups according to planned schedule is to make this cheap mainly on labor and equipment rental costs.
It is mainly a reliability thing, because the UL certification tests were performed so they "cover" the lamp safety performance only over the official life rating. So past that, the UL certification becomes invalid, which makes such lamp illegal (installation uses parts without valid safety certification, so Code violation) in most commercial settings (in home settings no one usually cares about code compliance).

Technical arguments could be:
Many MH lamps tend to rectify when failing past the end of their rated life and that tend to severely overheat the ballast if that is DC conductive on its output (so HX or a series choke). This is the reason why practically all European lamp makers require the use of thermal cutout equipped ballasts with all MHs.
With the typical US CWA, the rectification obviously can not overheat the ballast itself, but the higher voltage drop (present in one polarity) causes the ignitor to trigger, so the ignitor wears out rather quickly.
But probe lamps do not use ignitors, so this is not valit for probe start lamps on a CWA either.
The one problem remains is the elevated probability of the nonpassive failure mode. But this isn't that much unlikely even before the rated EOL, so a measure is required in the lamp operation to ensure the arctube explosion won't be that energetric during the whole life (shut it down to fully cool down before restarting at least once per week, so if the atrctube integrity gets compromised, it explode during the warmup, when the stresses in the quartz are the greatest and the pressure not that high yet), so the expired rated life is not that much related to this problem.
What remains is the efficacy deterioration, which makes the lamp not meeting the minimum legal performance requirements (put in place as a condition to allow their sale).
Plus the lamps color uses to be so way off.
So the truth is, it is very bad practice to let the lamps degrade past the rated life in normal service (so not talking about history exhibit or collection of those lamps).

Don't forget mainly the last two are the very frequent argument of "very bad efficiency of older technologies", put behind all the various lighting technology bans and those who were responsible for such mismaintenance become partly responsible for the bans existence - if the old technology installations were all properly maintained, so cleaned, lamps not let in place when degraded, the resulting low efficacy then gave the main argument for the greenbrainers to support the bans.

I'd replace them at rated life just to reset the severe lumen depreciation.