A raised planter in that area would look good, but that would cost as much, or more than having the stumps removed.

Fake grass will not be an improvement for this space. You don't need to remove the palm stumps, but you can cut them lower so they do not sit above the gravel.

Cut the stumps lower. Put in a better, taller block/paver edging to keep the gravel in the bed. Freshen the gravel. Buy some much larger decorative pots and fill them with tropical foliage plants of your choice to suit the light conditions of this space.

Go to your local garden center. They will be able to tell you how to get rid of the stumps....at least to below ground level. There are chemical products that will help to disintegrate the stumps further. As suggested above a good edging will greatly improve the area and keep whatever mulch.or stones inside the bed.

Why not just drill some holes in the stump and fill with stump rot they will eventually break down. Then extend the concrete to that area and use some nice big pots filled to overflowing with seaonal plantings IMO the gravel is not attractive and pulls you eye right to the area that should be really pretty you could even do something really simple like this if you don't want the bother of the other pic

I love your house color.

This gravel area should be a regular old planting bed and planted with plants!

I suspect the trees were removed long after the pool was installed. They probably looked good...and then got too big?

Artificial grass requires a lot of prep. You can't just cover stumps. And it'll look weird even if perfectly done.

I would remove the gravel (that has no place by a pool), improve the soil if needed, and plant to fill the space. Don't worry about the stumps--put a potted plant on them, use the stump rot stuff, cut them lower, grow something that will creep or drape over them, or ignore them. The big rocks are fun. I'd plant around them. Maybe add more. You can do a mix of things like a proper bed or all the same plant.

I don't know what grows where you are (probably everything!) but something easy might be agapanthus. They aren't messy, don't hurt when you brush against them, and can fill a space solid or form a relaxed hedge around a bed. Or a nice blend of what you already have in the pots. If you like mint, now you have a place to grow it where it can't get into everything! A spot on the windowless wall for bananas or giant bird of paradise where the wind won't rip up the leaves?

IMO you also need to soften the stark walls around the pool with some plantings and vines.