Once in a while I might have a fire if staying in a campground that allows it, never when wild camping. I think I had a fire once on the entire Southern Tier and never on the Pacific Coast or Santa Fe Trail. We did have fires now and then on the Trans America and the Sierra Cascades, It might have been 1 out of 10 nights at the most on those tours.

I wouldn't even consider carrying an axe, shovel, or saw on a bike tour. If I have a fire I burn stuff that I can break up or once in a while buy a bundle of firewood.

Not sure what you mean by "important part of sleeping outside". Do you actually keep a fire going at night while sleeping? I never do that when touring. I have mostly just had a fire in the evening before turning in for the night. I think it was only once I built a fire in the morning. That morning was 18 F and I wanted to enjoy the warmth of a fire for a while until the sun got high enough to warn things up a bit.

I find that the large majority of the time I manage to use indoor facilities or outhouses, but when necessary I dig a cat hole with a stick or tent stake. That is an infrequent occurrence though.