OLD TITLE: Smooth mesh to hide the low poly subdivisions
Title of the question changed, as the main issue seems to be that with the triangulate modifier and the subdivision modifier, there are new faces and vertices being created.
New question:
Using the Bevel Modifier, Triangulate Modifier, and the Subdivision Modifier, new vertices are being formed as you can see in the second image. I would like it to look like the first image, except with the Subdivision Surface modifier working, so the cutout loos smooth.
Beveling with no Limit only shows these two problems:
New Blend File:
Old question:
I created a cutout of a cylinder in my mesh. I beveled it, and it gives me this:
I would like to 'smooth' the edges to look like a nice curve, and not blocky.
I used a Subdivision Surface modifier, but:

I already tried going around using Ctrl + T, but I could not get rid of them all, and it made my mesh look messy.
Is there a better way for me to subdivide it without the vertices going nuts?
EDIT:
R.M's method below works up to an extent. It curves the cutout, but the face deforms appear still. Is this more of an issue with my mesh?
With no Bevel Limit, the corners are sharp at the front of the mesh, but the bottom still has the corners warping in. With the Angle, it makes the mesh more round/smoother, how I want it, but the bottom still has the mesh warps. As well as the front corners.









no limit. The only problem I'm noticing is that the round parts are not smooth, but you may have to go back to change this (use cylinders with more detail). – Reinis Mazeiks Jan 16 '18 at 00:14segmentsreduce the problem, but I don't think it can be eliminated without manually working with the mesh. – Reinis Mazeiks Jan 16 '18 at 00:27