Yeah as people above have said need the right kind of amp setup to power that many speakers at once without causing low impedance issues and running an amp beyond its specifications.
Another interesting alternative to sonos or a full on pa amp system is to look at options like Yamaha MusicCast. A unit like the rx-v679 can power two zones with 4 existing speakers and it is all controlled with a smart phone app to stream music and other external tv/blueray sources to multiple zones. Zones can be independently contolled and you can expand it with more than one AVR unit or their other MusicCast devices.
The RX-v679 may also be able to do up to 4-5 speakers in zone 1 and 2 speakers in zone 2 if you use 5 channel stereo sound mode in main zone. This may allow you to get away with just two of these AVR's but if you drive that many speakers from a single unit all going at once you can't go too high in volume!
The reason this solution is quite good is because it is so simple to setup without needing lots of custom devices, amps, speaker selectors etc. Just an AVR connected to speakers and your network.
You wrote that you have 9 speakers around the house, aside from the main room- are these single point stereo (dual voice coil and two tweeters) or single channel speakers? If they're single channel, are they being used in a stereo configuration (you would end up with an odd number of speakers connected to one channel) or was the receiver set to Mono, where each receives the same signal? If possible, a diagram of the system would help.
Can you list all of the equipment the new shop is recommending- is that $3000 with installation labor & setup?
Some AV receivers have additional zones of output, but operating these extra zones can be very clumsy and counter-intuitive. If you want to keep it simple and assuming the 9 speakers aren't presenting a load to the amplifier that makes it want to pack up and go home, I would recommend adding a good stereo amp that can handle a low impedance load, has the ability to turn on when the signal signal reaches it or turn on when it receives 12VDC. If you want to use Sonos or Denon's Heos (which many believe is the better sounding of the two), Yamaha's MusiCast, AirPlay or any other system like them, you will need a reliable WiFi network in order for this to work. Even in an area where WiFi signal wouldn't seen to be a problem, Sonos works better when at least one piece is wired and the others are able to communicate with that one (just saw this a few days ago).
Sonos and Heos have a device that acts as the source for this kind of system, without an amplifier or speakers- because you have more speakers than a simple pair or two pairs, I would suggest using one of these with a separate amplifier because whatever you send to the speaker selector is going to be shared by all of the speakers, so 100W in will be less than 10W to each speaker (because some is lost as heat). If the selector can't handle more than 100W, something else may be needed in the event that you want the music loud in those areas. Do you have a volume control for Speakers 1, 2, 3, etc on the selector? If so, you could use a multi-channel amplifier from Niles, Russound, Dayton (house brand for Parts Express and it works well). It can be 40W/channel in stereo, each pair can be set for bridged/mono at 80W, it can turn on with audio signal or 12V, it can send 12V to something that needs it (muting volume controls, another amp, etc) and it has two separate sets of input jacks, which can be set for separate zones or Bus. You can mix and match the stereo/mono designations, so outdoor speakers that require more power can get what they need without sending more to speakers that don't need it.
Assuming the speaker switch can handle the power, I would get the Heos Link or Sonos Connect, fed to the receiver as its own source and also to the input of a separate amp. Each has an Aux input, to accept the signal from your receiver, in the event that you want to listen to something that the Heos or Sonos can't provide, like a ball game, TV audio, turntable, CD/DVD/BD audio, etc.
A Heos Link, new receiver and a separate amplifier could be purchased for about $1000- $1500 if you don't start with the bottom or top of the line. I would be surprised if it takes more than 1-1/2 hours to set this up unless you need to run an ethernet cable to the equipment location.