Op you absolutely need cross boards/slats.

You should not use a flat plywood platform to accomplish the same outcome, it will be more expensive, less stable over time and contribute to eventual mold in your mattress unless you live in like Death Valley (and maybe even there too depending on how you humidify your house!)

Again, you without any qualification need to use slats or cross boards to support the mattress and will be unhappy if you attempt to use ply for that.

If you want the “clean look” of a plywood platform, use an overlay of decoratively cut (50%+ negative space) 1/4” wallboard (freedom ply here in the us!) on top of your slats. When someone asks for a plywood platform over their slats, which has happened twice now, I usually set the saw in a jig and rip a bunch of four or five inch strips off some 1/4” and put them an inch or so farther apart than their width (so a bunch of 5” strips would be 6 + inches apart each) and join the whole thing together with a thinner strip across the top and bottom underneath held together with countersunk screws that have either acorn nuts on their bottoms or something else. You need countersunk so stuff can not get caught on the top and acorn nuts on the bottom so hands and cats don’t catch any strays fishing around under that thang. You can’t use glue because you need the whole apparatus to go parallelogram when the bed frame does.

E: oh yeah, if you can tolerate it, check the dump/your local listings for a busted ass metal frame with rails long enough for your design (or close enough not to matter). Those steel L brackets are a super good way to make sure the wood doesn’t pop apart over time. Use some kind of a poly product where wood supports metal to make it even better.

E2: reclaimed metal parts look really sick when you sand and repaint them a loud color that no one will see until they take the mattress off.

E3: if you bake enough slop into your overlay strips you can actually glue them in the case that you trust yourself to glue square and true. Okay I’m really done this time.