But we're going to go a step further, as well. Warning, the following is super techy. You do not need to understand it, it's just for the theorycrafters.
Conceptually, the intended difference between random chance procs and On-Use effects should be that the chance ones are unreliable, whereas the On-Use ones are active when you want them to. This extra proc chance on the pull (which you identified is actually stronger than intended) goes against that rather significantly, ensuring that the random chance ones are active when you most want them to, on the pull.
As such, we're going to change the functionality of RPPM procs in Legion. The TimeSinceLastProcSuccess and TimeSinceLastProcAttempt will both be set to 0 on the start of boss encounters, instead of the 120 and 10 respectively that it currently is. This will significantly affect the overall DPS value of these trinkets, and we'll adjust tuning on all trinkets to account for this change.
I think the issue behind why the current system was originally implemented is that there are two brands of "random" trinket currently - RPPM and % chance to proc + ICD. The issue is the % chance to proc + ICD variety is actually incredibly consistent (pretty much get a proc upon expiration of each ICD window - and these trinkets also start boss encounters with their ICD window cooled down, so you can reliably get a proc on the pull with these trinkets, especially as a dual-wielder).
Are there any changes to this flavor of trinket being planned, or "tuning will fix it?"
Yes. ICD trinkets are relatively rare these days, but are a different flavor that we'll still use from time to time. As others have pointed out the extra proc chance on the pull for RPPM trinkets was significantly related to trying to compete with those ICD trinkets, which were the vast majority of trinkets at the time. Nowadays, they aren't the norm, so we don't need to make RPPM ones overpowered to compete with them. ICD trinkets will likely be a little less consistent, but still more consistent than RPPM. We don't want to have too much difference between ICD and RPPM, because it's a distinction that the vast majority of the playerbase shouldn't need to understand.
If you want consistency, On-Use trinkets will be attractive to you. If you don't mind randomness, chance to trigger ones may be up your alley.
Overall, we're attempting to cut down the on-the-pull cooldown stacking, and this is just one angle that we're attacking that problem from.