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I suggest you contact your local extension office (found under gov't in the phone book) for help with this problem. Since it appears this mature tree has just started throwing seedlings, there may be some underlying problem that isn't readily apparent. Something is going on that's causing the tree to try to reproduce itself, hence the abundance of seedlings. Pulling them by hand seems the only way to control the problem at this point.

These articles should help: https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm
https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-pruning.htm

This could be the onset of a fungal issue or insect pests. I would recommend treating the tree with neem oil, which will take care of both problems. Here is more information: https://www.gardeningknowhow.com/problems/pests/neem-oil-uses.htm

I would certainly take into consideration the trees that the cemetery recommends. A flowering Plum or Kousa Dogwood would be flowering trees.

Also consider Service Berry, River Birch and or Blue Beech trees.

I would encourage you to visit your local garden center or tree nursery and discuss the planting site and restrictions. They will be able to help you choose a tree suitable to your growing zone.

Unfortunately, there is not much to be done or that can be done. You can try wettable sulfur in the soil to keep it at bay, but over time it is likely to return if they are not cut off properly.

This looks like the beginnings of black knot fungal infection. Here is more information: https://www.gardeningknowhow.com/edible/fruits/plum/black-knot-tree-disease.htm