You are absolutely right that pilots tend to do many of the ancillary duties, like cleaning or scheduling, in addition to flying. In general, you will want to seek out companies that have many helicopters, because companies with just one aircraft will usually have the pilots doing those types of duties. I have tried to come up with some good ideas, but there aren't too many.
Many of the larger flight training schools employ one or two people to handle scheduling, enrollment, accounting, advertising, etc. So, they basically run the office. Here is an example:
As you can see, they have "operations managers" or "student services managers" in addition to pilots and mechanics. Helicopter charter companies will require staff to handle scheduling. Other helicopter operators with a large fleet will require similar support staff as well.
Another possibility would be to work as a dispatcher. This job does require some training, but it would be related to aviation. Here is one site, but you can Google "aircraft dispatcher" for many more.
There are some companies out there that do detail work on the interiors of aircraft. They clean them and install new upholstery, etc. They also paint the exteriors. I don't know which companies do that, or what training they require.
You could work as a ground instructor to train student pilots. This would require some training as well, but not nearly as much as becoming a pilot. You would have to take classes and pass written exams, but then you could give the lectures to students at a flight school.
Lastly, I have worked with some people who specialize in aerial photography/video. They charter a helicopter and take pictures for their clients. This would be technical on the photography side, and it may be expensive to set up, but you would get to fly around in a helicopter to take the pictures.
I hope this gives you some additional ideas. You may have to get creative to find exactly the job that works for you. Good luck!