1. nothing removes nitrites and nitrates except for water changes. Dirt provides nutrients where sand does not but dirt floats so in order to keep it down, you use sand over top.
2. Tanks should have a water change once a week. If you dig into the sand when you use the gravel vacuum then you're going to suck it all up. It's not like gravel where it will go right back down after a few shakes. Sand is light, dirt is even lighter.
With the weekly water change I always clean into the gravel or in my sand tanks, skim over the top of the sand to get the poop and debris. You can't just do it "when it looks yucky" ideally you should be doing this every week. Tanks don't clean themselves, unfortunately :-/
And yes your fish can be in there when you clean as long as you're not doing a 100% lol
So you have bare bottom now, do you use a gravel vacuum now? Or do you use a different method?