3 pictures and I still feel like I don't know what's going on.

Do you see what looks like wood falling off the building right at ground level?

It doesn't appear that "board" is falling off the house - as in it is supposed to be an integral part of the structure and is needed, but in disrepair. It just looks like a thin fiberboard(?) someone used to cover up what looks like a mess of housewrap for the stucco drainage plane that is sticking down or some other plastic sheeting they have stuck against the slab, all of which was just oversprayed during the paint job. The "board" looks to be just pushed up against the house by the grade, not fastened.

Then there looks to be another landscape edge strip 8" away holding back the higher grade - that one might be recycled rubber though. Really there should be no wood in contact with the ground, especially at or on the house.

There is no issue with the landscape border in the ground. That can be ignored.

This is all about what’s connected to the building.

Of course, I can see the board is not structural and is not doing anything to hold up the wall. It is for aesthetics to keep the wrap behind it out of view and maybe to help keep water out. That’s why I was calling it a trim piece.

It used to be firmly connected, but is falling off over time. It is also broken off at one end as shown in the first photo.

Since it‘s partially coming off anyway and looks ugly, I don’t want to do a patch job to fix it. I want to rip it completely off and start over with something new that looks like it’s ”supposed to be there“ and will not fall off again. Then it will get painted over when the house is repainted this summer.

Any ideas of what would be best used to replace whatever that is with something better suited to last a long time and not lose adhesion?

Well you certainly expounded on your comments that should've been included in the OP. The best, and proper detail, would be exposed concrete of the foundation, 4"-8" above grade. There is no reason to protect the concrete from water. The stucco drainage plane should stop at the weep screed. Can't tell what else is going on if there's additional plastic sheeting, but it shouldn't be there if so.

Even if it doesn’t need to be there to protect the concrete from soaking in water, I want something there even if it’s just for looks and is only a ”trim piece” that isn’t performing any important function.

It looks very rough and unfinished behnd the board. I still want something to replace it even if it’s purely cosmetic.

If it were my house, I would be cutting off the WRB/plastic (or figuring out what the extra plastic is - pics are hard to interpret especially when its all painted).

However, if you must have a "trim," make sure that it does not touch the underside of the stucco weep screed. And, it sure looks like it's WRB/plastic sheeting that's folded/rolled/airpockets goes up to the weep or maybe behind the stucco. So then 2 things, any adhesive will not continue to hold a "trim" against the plastic. And you want to prevent capillary action, so that is why you keep it down from the weep screed and if the plastic goes up, then the "trim" may allow potential capillary action behind the "trim." Water going up the wall is bad, as well as attracting and having a highway for pests, like termites (which is why codes dictate exposed concrete of 4"-8").

So, if you continue with the "trim," to avoid wood (note composite decking like Trex has organic material), you can use PVC boards/decking. However, it might be too thick (and expensive). Try looking at garage door jamb/header vinyl pieces. There are a few profiles in browns with different thicknesses available (e.g. jamb vs weatherstrip) - I have altered and used them for other purposes before. Good luck!