The electrical circuit is still there even if the device is off... I'm not an expert in this stuff, though, either in the electronics or the mixer side.
For mixers or switchers, switchers are simpler and potentially cheaper, mixers have the advantage of letting you have one or more inputs active or connected at the same time without ever having to manually switch, as well as giving you per-input volume control. They cost more, though. Still, entry-level mixers that support at least two stereo inputs start quite low. A typical 5-input mixer is going to be one mono mic input and two stereo inputs (usually labeled as channels 2/3 and 4/5), and maybe a tape input that can serve as a third stereo input in a pinch (with no specific volume control other than main mix, and expecting line-level input).
Behringer products have a reputation for being lower quality, but they're still better than noname Chinese products, and their 5-input mixer starts at $39 USD on Amazon.
https://www.amazon.com/Behringer-502-Pr ... B000J5UEGQ
Mackie has a much better reputation for being quality products (professional sound engineers won't laugh at you for using one like they would for Behringer), their 5-input mixer starts at $60 USD on Amazon.
https://www.amazon.com/Mackie-Mix-Mix5- ... B00ND1K8T6
Soundcraft has a reputation as being a bit higher-end than Mackie, theirs starts at $109 USD:
https://www.amazon.com/Soundcraft-Notep ... B075SK443L
There are many other mixer brands, of course. I'd say avoid Pyle, and avoid the many random or unnamed Chinese brands. Personally, I'd probably go for the Mackie, as it's kind of a nice middle ground of having a better reputation/quality than Behringer without being too expensive, but if you're in a pinch and need to scrape and scrimp, Behringer stuff isn't *that* bad and it's really hard to beat that $39 price.
Alternatively, look at a store that focuses on audio/video gear like B&H, and take a look at their utility mixer section sorted by price:
https://www.bhphotovideo.com/c/products ... OW_TO_HIGH
You can look at reviews there to see where there are complaints about noise/hum. Pay attention to if inputs are mono or stereo, as an example that first Behringer mixer says 4-input but they're 4 mono inputs, while some later also say 4 channel but are 4 stereo inputs.
You can do the same in the full mixer category, filtered here for 6 inputs and less:
https://www.bhphotovideo.com/c/products ... %3Aup-to-6
There's a 6-channel Yamaha listed there for $100 that's worth a look, Yahama stuff is good, and it can combine the first two channels to act as a true 3-input stereo mixer. And if you ever want to repurpose it for something else, two mic inputs both with 48v phantom power means it's well suited for light streaming/podcast work.
Or just buy a cheap manual pushbutton audio or composite video switch with RCA inputs. They can be really cheap (like $13 for a 4-input), and there's probably nothing terribly wrong with them since they're mechanical devices with no power...