Roundabouts

Yes.

1) Originally, I think that they made connecting all the "roads to cities/towns { x_j }" easy.

Before any grid-type system, most towns were seemingly built on a hub-and-spoke system, with the central square at the middle and roads heading out from there to the next towns, which could be in ANY direction.

Even after grids became more common in towns, the shortest path to "next towns" were straight lines to those towns, modulo deviations for better terrain, oxen/horses and carts, etc.

This leads to all sorts of rural intersections where roads meet at ANY angles. Turning the crossroads into a roundabout makes for a way to have a form of uniformity regardless of the angles.

The crucial difference in the UK v. where I am in the US is this:

the UK always has a sign with a "plan view" of the intersection, like you'd see on a map, showing the road names, the destinations along them, and the appropriate angles between them

That makes navigating them much easier. At some point, the roundabouts seem to have been starting points for new paths between them and other towns.

2) With automobiles, all else being equal, safety is worst when autos cross each others' paths; i.e., at intersections where one turns across other lanes of traffic. Protected turn lanes (with green arrows in the US) can help, but that induces waits for a lengthened signal program.

So roundabouts give a number of other advantages:

A) Traffic only flows in one direction, so they tend to be safer than what many view as the "standard intersections" with which they grew up--fewer accidents is good.

B) Traffic lights require programming. This programming is hard to get right for all traffic situations seen at an intersection AND inevitably introduces delays or more sensors (and sensors can be fickle and they depend upon the composition of the vehicle!). So roundabouts save

i) The cost of the lights, the supports to suspend them, and the ongoing installation and maintenance.

ii) No lights overhead means they can't become bombs in a weather event or line failure / accident involving supporting poles, etc.

iii) The programming of the lights, which may need frequent tuning, another potential initial cost and maintenance cost.

iv) The cost of sensors, including maintenance.

v) The cost of electricity for the above.

vi) Requiring that vehicles have enough magnetic material that they can trip a sensor, thus limiting advances in automotive construction.

vii) Adding new roads is relatively simple, with new signs being the only required element beyond extending the roads to intersect the roundabout.