It's not so easy. You seem to be under the assumption that there will be just one static system prompt doing all the work that you can customise to your needs. This may be true for some apps but many useful apps will usually do a bit heavier lifting.
Agree 100%. I've found myself avoiding most GPT-based chatbots for this same reason. I don't want it to be subtly manipulating things without my knowledge based on custom instructions that I don't know about. Adding a "view source" option would make this feature from "meh" to "worth the money just by itself" for me. I've been considering cancelling GPT Plus since I find myself using Kagi a majority of the time anyway, but that sort of change would keep me subscribing.
Meta note: This is one of the best posts I've read in a long time. Outstanding work!