Background
While various items such as maintenance and ease of tire removal are cited for the SSSA (Single Sided Swing Arm) design, initial testing and development of all these designs were started on the racetrack. Honda initially released their version of the SSSA with NSR250R.
image of an NSR250R

All of the early SSSA's were developed for racing to reduce un-sprung weight. From there, street sport bikes were the early adopters, porting the race technology to the consumer. This was followed by SSSA's on scooters and other non-sporting motorcycles.
While there is logic regarding reduced maintenance and ease of maintenance regarding SSSA's their initial adoption was for reduced weight and racing applications.
Un-sprung Weight
With all vehicles, un-sprung weight is the holy grail of handling. Un-sprung weight reductions result in a wheel getting back to the surface area it's adhering to in less time. The result is frequently less over-steer in cars and in motorcycles, less low sides due to interruptions in friction adhesion to the road surface.
Un-sprung weight or un-sprung mass is the weight that is not carried on the spring in your suspension. If it's below your spring it falls under this category. For bikes; the rims, tires, rotors, brakes, lower fork legs and swingarm all fall under the category of un-sprung mass.
The lighter this mass, the more responsive it can be to outside forces, responding as needed by absorbing energy, responding to undulations in the surface by not bouncing off the undulation and getting back to the surface of the road with it's full weight as quickly as possible.
To summarize the advantages:
Less weight creates less momentum when responding to a bump or undulation in the road surface. More momentum in response to an undulation could result in the tire getting bounced off the surface of road.
Less weight allows for the spring to push the unsprung weight back to the road surface quicker allowing for the contact patch to get back to work with friction for adhesion.
Why don't all manufacturers do it?
This is subjective, I believe it comes down to cost. If it were cheaper and there were performance advantages then I would imagine that everyone would engage in this design as a framework moving forward. It would seem there is a higher cost as only the higher end motorcycles like Triumph and Ducati are implementing.
The flip side is that due to lightweight materials and construction other manufacturers have found a weight/rigidity balance in there designs that allows them to be both competitive in the showroom as well as the racetrack.
Scooters
Again, this is subjective and opinion based.
Scooter swingarms are inclusive of additional properties other than rigidity for the rear wheel. Other properties of a standard scooter swingarms may include the following
- Centrifugal clutch
- Drive Belt tensioner
- Shaft drive
- Secondary shaft sprocket
- Rear Brake
Here is an example diagram of Honda Ruckus that does not include the rear brake mechanisms within the unit.

Due to the rigidity of the unit it only makes sense to make it all inclusive in a single sided swingarm. The unit provides multiple options for the various components as well as ease of maintenance for mechanics.
Additional benefits include ease of rear tire change for an end user that may not have the technical ability to line up a rear wheel properly with a traditional swingarm making for less service related mishaps and longevity of the vehicle.
Errata
Although car related, this kid has some excellent videos, a little long winded but very good information regarding suspension and the importance of a low unsprung mass in any vehicle. Have a look here.
Many manufacturers use a vacuum casting method to create thin walled high strength components. This a good video that illustrates the benefits.