Is SDR technology a way for me to listen to a large group of stations or even the entire FM band at once?
Yes. To actually do it, you need:
- SDR receiver hardware with (slightly more than) 20 MHz bandwidth to capture the entire FM broadcast band.
- A computer, fairly powerful (definitely not a Raspberry Pi or such).
SDR software which can be configured to actually do this. Many receiver programs are focused on receiving a single signal; my own ShinySDR can do multiple but not as efficiently as a purpose-built program.
The likely place to start to make this actually happen would be to use GNU Radio, a framework for making SDR applications. GNU Radio's polyphase channelizer can efficiently separate all the FM channels since they are conveniently equally spaced. You will also need something to do with the hundred separate channels other than playing them all at once (unless that mess is what you wanted).