Here's a take on the question which includes
I suggest that it is time to acknowledge this reality and reinstate the opportunity for petitionary prayer in the Shabbat Amidah.
Knowing that the Rabbis frowned on such prayers didn’t stop our ancestors from saying the “mi shebeirach” prayer for healing on Shabbat. They even inserted a disclaimer in the prayer, the words “Shabbat hi milizok,” which can be loosely translated as “we know that we’re not supposed to cry out to You on Shabbat, but we’re going to do it anyways....” So there is precedent in the tradition for my suggestion.
Please note that I have no opinion about whether the Amidah should be changed or what the status of individual petitionary prayer should be or whether not doing it on the Sabbath contributes to the quality of the day. However, I'm charmed by there being a word for "we know there's a rule, but we're going to break it anyway". A stiff-necked people, indeed.
I try to forgive other people's ignorance, considering that I can't even get the details right about what is either my own culture or at least the culture next door.