No one seems to service the pivot pin/shaft for the brake so some of them become really sloppy.
OK, an exageration....two of you have serviced yours but likely not more.
Depends on how sloppy it is but if the pedal moves excessively while being tight to the spline. You can bush the mouting to remove the clearance. I've bushed quite a few brake pedal pivots on various bikes, using copper tubing or brass bushing stock from an alternator/starter repair shop or automotive parts store. The fit doesn't need to be precision so one does not require reaming to machine tolerances.
You might try some nylon tubing as that can last a long time if a push fit is used.
Before doing anything, I recommend that you check another KLR, try the dealer's new stock as the brake pedal isn't quite as close tolerance as the camshafts.
Regardless, put a pedal service on your yearly list.
FWIW, we looked at adding a grease nipple when we were doing rear suspension grease nipples but concludes that the bushing length made it impractical. Seems better to just pull apart, clean and use a good quality general purpose automotive chassis grease.