We just built a new home and set it up with wired speakers in 4 other rooms all running off the main AV system in the living room. I spent the big bucks and had it done professionally but here are a few comments and suggestions from my experience.
1) Evaluate if you really need the capability of playing a different source in different rooms at the same time. That will raise the complexity and cost of your setup exponentially. The main thing I wanted was to be able to hear TV audio or whatever music was playing on the main system throughout the house. For example, I like to hear music in the laundry room as I fold the laundry. I don't need to hear something different.
2) I'm assuming that the builder installed volume controls for each room with speakers. It is critical to be able to control the volume in each room independently.
3) You will need a multichannel amplifier to feed the rooms with the speakers. There are several on the market. I went with an Anthem setup and it works great.
4) I'm not sure how you will arrange to get the audio source from the main living area to the basement rack. It's important that you can easily change the source from your main living area instead of going down to the basement everytime you want to play something different. For example, you could use a streamer that is hooked up to your local network and control the streamer with its APP anywhere in the house. I have an Eversolo DMP-8 hooked into my Anthem A/V receiver which allows me to listen to the audio from the TV or audio on Qobuz anywhere in the house. I can control Qobuz from my phone by using the Eversolo APP that lets me control it remotely. If the game is on the TV and I'm in the kitchen I can hear what's happening. Or if I'm playing music through the Eversolo I can hear that as well.
5) If you are going to add speakers to rooms that have been wired you need to ask your builder for details on how the wires were placed in the ceiling and where they were located. Putting in ceiling speakers is a lot harder than you would think. I went with Paradigm speakers and they sound great but they were expensive little buggers.
Good luck. That "insane" amount of money you were quoted may sound more reasonable after you get into this project.