When is a control tower required?
As far as I'm aware there's no specific regulation that requires a control tower to be established. FAR 170 and the regs referenced therein set out when an airport becomes eligible for a tower, but nothing requires an eligible field to have a control tower (in fact the FAA won't establish a tower unless someone gives them the land).

That said it would probably be a Bad Ideatm for certain airports to operate without a control tower (Atlanta, Kennedy, O'Hare, LAX...) - in fact those airports are busy enough that they have additional traffic management procedures in place beyond what a control tower can provide.



Control towers exist to improve safety (improve risk) as you alluded to. There are procedures and rules established to allow pilots to self-sequence at untowered fields (you'll find those in section 4-1-9 of the AIM), and they work very well when everyone follows them.
The catch is everybody doesn't follow those procedures - or more accurately they're followed with varying degrees of precision (my personal favorite example of sloppy procedure is when you hear someone say "any traffic in the area please advise" at an uncontrolled field). These procedures also don't cover other movements on the airport surface very well: Some pilots will announce their taxi plan at an uncontrolled field, but there are plenty of pilots who taxi from point A to point B with their head up and their mouth shut so as not to tie up the frequency for aircraft in the air.

As the number of operations increases at an airport the various degrees of precision and adherence to the uncontrolled-field procedures in the AIM  becomes more of an issue, and a control tower may be established by the FAA or the local airport authority to safely and efficiently coordinate aircraft movements.