In a properly cycled tank you're BB will process out the ammonia. Carbon is really only needed if you are removing meds or odor from the tank. It's really a personal choice to spend the money on carbon as you will have to replace it every 4-6 wks.
It's absolutely fine not to use carbon. I have removed it from all of my filters because I simply cannot be bothered to replace it all the time. I fill the gap with sponge, which is a great method of filtration as it has so much surface area for BB to grow on. :)
I do the same. The only tank with carbon was the snail tank. The water would reek like a day after a water change without it. Every now and then I will stick an actual cartridge in there but in general - its all filter foam/sponge for aquaclear filters.
In betta tanks it's a matter of choice. I have several tanks running on sponge filters which have no carbon. I also have a marineland hex 5 that has carbon in the filters. Tanks this size don't tend to get the odors bigger tanks do due to the diversity of inhabitants in bigger tanks. I would not run my 75 or my 30 without carbon.