The parish where I currently attend to our construction woes seemingly orders the prayers to complement the lectionary. I forgot about these. We had them in my home parish too. I don’t absolutely love these: nothing is objectively bad, and they go in hierarchical order, but they don’t always have explicit or even implicit prayers for the whole church on earth, for the pope and bishop, for the clergy etc. before moving to other intentions. I don’t think that this would merely repeat what is in the anaphora, since that doesn’t include the other faithful (the lower orders of clergy, virgins, religious, etc.) and can be tied each week to a specific intention.

The bishop’s office requires the missal prayers for confirmations. I believe that this solves the problem too.

As someone who had responsibility for composing these some decades ago, and then upon further recollection and contemplation realized gaps in my own daily prayers of general intercession, I would like to suggest for consideration adding a prayer between those for the sick and those for the faithful departed (the latter should be included but there is the more important prayer in the Eucharistic Prayer for them), something like this:

Let us pray for your servants who are dying, who may have lost the will to live, or who may die today or in the coming days; Lord, grant them the mercies and grace of dying the happy death of the just according to your gracious will.

We neglect those categories (the may overlap, but they are also distinct) of people in our pubic prayers, and I believe we strain to retrain ourselves to be mindful every day of them. Where they go, we too shall follow: the most basic affirmation of common solidarity.

I have another general prayer of daily thanksgiving that can be adapted for a more general intercession:

Let us pray: Lord, stir up within us the spirit of thanksgiving for all the blessings and prayers you have answered according to your gracious will, especially those of which we know not, that we do not understand, that we do not appreciate, or that we take for granted.

My personal private version of that adaptation is simpler: