There are many types of frames for paintings. It's true for the classical painting frame that it overlaps, and I don't think it will increase as the size of the painting increases.
With mats you face the exact same problem: mats won't cover part of your painting in addition to the frame - they usually just have rabbets of their own in which the artwork is placed.
The problem is that the overlap is necessary for the frame to hold the artwork.
The width or depth of the recess in the frame or overlap of the mat is dependent on the character of the artwork's edges, and must be at least as wide as the largest deviation from a straight line, or you'll end up with gaps between the frame/mat and the canvas.
Taking that into account, I suggest getting a floating frame:
From the same page were I got the (first, edited) picture:
Floater (or “float”) frames are so named because the canvas appears to float inside the molding. Their biggest benefit is that they leave the canvas edge exposed rather than covered by a rabbet. Another advantage of the floater, no matter how narrow, is that it helps keep the canvas from warping.
Additionally, as you can see, they're easy to make yourself, and you can adapt the thickness, colour, wood type, grain, &c. to what best suits your work.
Because of the open design, they won't keep dust away, but you can of course insert a glass panel in front and seal the backside off if you'd want to (which might making one a lot harder, though).
You can also contact a frame-maker and ask for their opinion. Many art supply stores also offer custom framing services.