In a good brew I thought there isn't supposed to be too much yeast left.
Actually this is only true in the wizz water sold by most commercial breweries. They have convinced us that their flavorless, thin, yellow water is the standard for beer. This is why they insist their beers are to be served so cold--shock your tastebuds so they can't discern how bad the beer really is.
In any homebrew you should have small amounts of yeast living and working all the way up until you drink it. This is part of the reason to decant homebrew (pour into a glass) to leave the yeast residue behind.
Your beer will be naturally carbonated by the action of the yeast on the priming sugar (or malt extract) added before bottling.
As for the purpose of secondary fermentation. It produces little in the way of alcoholic strength and the carbonation bleeds out through the air lock. The purpose is to refine the tastes of the beer. It will allow the yeast to work on the beer and improve the flavor.
I have made batches with identical ingredients and techniques, but put one through a secondary fermentation and bottled another right after initial fermentation, the difference was quite pronounced. You will produce a much better brew if you give it chance to secondary ferment.