There are more differences in a true .223 Rem vs. a true 5.56 NATO round then pressure. Modern rifles have chambers cut to safely fire both in most cases (certain countries forbid rifles in a military caliber so, they make true .222 or .223 Remington caliber models) for those markets.
Pumps in general seem to work best in "snap" shots. They can be used for long range and prone shots but, they aren't the best option. How many centerfire rifle caliber options do you find in the marketplace today? 22lr gallery guns are really a different category and don't rely on any recoil to facilitate cycling the action.
Lever actions are the ones I have a hard time understanding since most are really built to be carried in a scabbard on a horse and not slung on your shoulder. There are a few modern versions which are very nice but, most models sold today are based on the classic Winchester 94 or Marlin 336. 20 years ago you could buy one in 30-30 pretty cheap compared to bolt actions but, these days a good bolt action costs less and works better for most applications.
Bullet selection is also a problem for any tube loaded rifle. High drag round nose bullets don't work as well as modern ones when the ranges extend. Yes, you can shoot a 30-30 or 45-70 at the 600 yard line at a DCM meet but, a higher velocity ballistically efficient bullet will work much better. I think that's one reason Browing made the BLR rifle which was magazine loaded, not tube fed.
If you factor malfunction clearance drills into the mix, tube fed rifles are a real pain. Pumps and lever actions are harder to clear then a bolt action. Most people aren't in a police force or military unit but, they want the perception of easier malfunction clearances in a SHTF scenario (which is false security if you don't practice malfunction drills).