I used the Zoom H1 for 10 years. At gain level 37 (+ 13 dB) - not more, not less - it has slightly better EIN than better handheld recorders like the Roland R-05. I used it along with a Church Audio preamp to power my mics AND add gain/attenuate the signal depending on how loud the sound source was. It served me very well, but you're right that if you need the recorder to supply more gain than +13 dB, you start noticing the pre-amp self-noise. And if you're dealing with high dynamic range content (i.e. you have to dial the gain down so the loudest parts don't clip, but there are still quiet passages), then none of these 24-bit recorders will give you clean pre-amps for the quiet parts without proper gain staging, even the ones that are more well-regarded than the Zoom H1.
My rule of thumb was that I'd set the gain at -10db for what I imagined the loudest parts would be (and the highest peaks would end up at -5 dB or so). Most of the music would range from -15db to -25db, and everything would sound absolutely fine. If any of the music was quiet enough to hover around -35 dB, that's when the preamp noise would start getting noticeable.
I assume the same is true of the Hx Essential line because the preamps are the same. Good for loud music, not so good if the music gets quiet.
I've been using the F3 with a Uši phantom adapter for a few months. The only downside is battery life, but I've been getting ~6 hours on 2 rechargeable AA batteries by setting the phantom power to 24V (which then gets converted to 8V by the adapter). I'm getting much better performance during the tricky passages I explained above, but I'd say they were only ever present in about 20% of the shows I ever recorded.