What follows is based on this article (HTML and PDF), lots of research and more time than I like to think about wandering around in the boonies, wondering where the heck I was.
Kilcullen has a similar article posted around these parts that addresses the phenomena of smaller assault units and larger suppressive elements in an attack, especially urban.

A group this size is capable of maneuver, suppression and assault, while being able to communicate and lay down relatively heavy fire support in the short term, pending a call for proper artillery.

At the same time, this squad is able to absorb casualties and still function effectively, since it would have to take c.60% casualties to seriously cut into its ability.
It's interesting that you say that, because I'd betcha that once you build to a certain size and decide that it is appropriate for a set task, it won't be able to operate at that level of efficiency once you start taking casualties...because you have all of those 20 men tasked out. This is just another way of saying that a 13-man squad could have performed the same task, IMO.

The size also adds to the security and policing mission, by having more "boots on the ground" for presence.
This is another interesting statement. I understand where you are trying to go with this, but "presence patrolling" is not a tactical task. If you need a factor of twenty-something infantrymen, just employ two 13-man squads and retain a third in QRF role, for rearm-refit, etc.

3. Three of these squads to a company, plus a full sniper platoon of 20 (8x2-man teams, plus PltLdr, PltSgt, 2xRTO), and a 20-man HQ Platoon (including a 10-man mortar section).

4. Seven of these companies to a battalion, plus HQ, Support (MT/Maint/Supply) and Weapons companies of 100 each.

This battalion organization puts seven maneuver elements in the field, vs the three we typically have, now.
Why get stuck on companies as the lowest unit capable of maneuver? The tripod of maneuver elements exists because time and studies have proven that the number three is at the upper limit of what a leader can exercise command and control over. Sure, the study needs to be updated with consideration to the tools we currently use for command and control, but we'd still be pretty hard pressed to find that the planning factor has become irrelevant.

And snipers organzied at the company-level. Ugh...No way jose, methinks. We have a hard enough time effectively employing them at the battalion level as it is.