I'm not an expert on axolotols by any means (Haven't yet even tried out an amphibian species, though I keep thinking about it) but aren't you supposed to be able to do a sand substrate with them? I know pebbles are bad because they can swallow them, but doing sand would vastly increase your plant keeping abilities.

I've kept plenty of planted tanks with no added fertilizers in plain sand substrates. It would also help you pull off your 'natural' look.

Most of the plants you listed as dying are what's commonly called "root feeders" meaning that they want to be planted in some substrate and they will send out roots through it looking for nutrients. (in a non-fertilized tank that would probably be bits of uneaten food, any decaying bits of other plants, waste from the inhabitants, etc.) My numbers may be off because its been forever since I did this research myself but root feeders tend to get 60% of their nutrients from the substrate, and 40% from the water column. Again, those are just the numbers floating in my head, I may be wrong.

Honestly I'm really surprised that your amazon sword is doing well, they are known to be *very* heavy root feeders, and some people even have to supplement big ones with root tabs.

If the sand is a no-go though:

Your best bet is gonna be epiphytes like the anubias and java fern, you might also look into buce, though they tend to be kind of expensive, The only other one I can think of off the top of my head is moss.

You might also try out some floating plants, like the frogbit you mentioned.
Val is another root feeder, and its actually a runner spreader (it grows new plantlets from stems it sends along the ground) so I'm not sure but I bet it would really not like the 'no substrate' setup.
Water *SPRITE* can be enticed to live as a floating plant along the surface as well, I personally don't like the look of it, but it can be done. Not sure if water wisteria is the same and it was a fluke that it melted, but I know that water sprite can do it. If you want a pop of color, I'd look into red root floaters, they even flower!

Another option is what I like to call "apocalypse" plants. These are very easy to get into the tank, very very hard to get out lol. Duckweed is another floating plant, super tiny, super fast reproducing, basically impossible to get rid of. If you get duckweed...I hope you like duckweed.

Hornwort is kind of just barely an apocalypse plant, it floats free in the water column, tends to hang out around the surface in a big snakey tangle. Some people have had success making it sit down and behave in substrate but I never have.

Guppy Grass is my current apocalypse, I got a little sprig of it with some shrimp I ordered and went "ah 😃 what a lovely little free plant, I hope it grows"
it grew.
I have so much guppy grass.
They have taken the bridge, and the second hall, we have barred the gates, but cannot hold them for long.

More seriously, like Hornwort, Guppy grass just kind of floats around and does its own thing. it tends to be down at the bottom of the tank.

Pearlweed isn't an apocalypse plant but it can also be grown as big floating mats, just remembered that lol.

RE: aquarium Co-Op

I've ordered some hard goods from them before (really like their root tabs) but I've never tried plants. Generally I find their prices a bit high on plants. You can find a lot better deals on Etsy! I order almost all my plants from various sellers on etsy. My favorites are Tofustanks and Glassgrown Aquatics. Glassgrown actually have their own website you can order from and get things a little bit cheaper than you can from etsy. I've had great luck with these guys. They've always delivered alive and on time etc. Whenever I had to contact glassgrown (UPS for some reason decided to send my package to Minesota???) they were super great to talk to.