I hope it helps. I have some palm tree seeds here that I can't get to sprout yet. If you feel like sharing your method, I'd love to hear about it.
Have you considered a parlor palm (Chamaedorea elegans?) You don't really get the trunk part of the look, (unless you're rich and buy a really big old expensive plant,) but definitely palm leaves. I've had a clump of this for over 20 years that was with me in OH for a long time, then the past 6 years here in AL. As far as palms go, I'm pretty sure this species is the most popular and numerous. That would be because they are so much less sensitive to dry house air while inside for the winter. Many palms can be devastated by that, which may be part of the difficulty your cute little sprout is having. As long as a new leaf is coming in, it's still growing.
Of course pics can be deceiving, but the soil looks very moist, soggy. Potted plants in a soil of all/mostly tiny/fine particles generally need to dry significantly between waterings or the roots can rot from lack of air combined with excess sogginess at the bottom of the pot. If there are organic particles in the soil, over time these will decompose into even smaller particles and literally be mud at the bottom of the pot. Very few plants can look great in this condition, and this is the primary cause of death for potted plants, beyond not watering it at all or leaving it out in the cold. Even after decades I'm often surprised when I repot plants because the surface has always looked the same, but what has gone on inside the pot is a complete transformation from fluffy "potting mix" into muck.