No, you can use others if you wish. But mixing boards for other applications may lack features that are useful when shooting film or video and you might end up paying for other features that you'll never need. For example, battery power isn't very common in boards made for mixing live sound yet it is essential for location film/video work. Most live sound oriented boards lack the tone generator important for aligning a mixer with a camera or recorder. Monitor inputs for a camera return are often missing. OTOH, a lot of mixing boards have 'special effects' sections with reverb and the like that you absolutely would never want to use for field recording applications.
Some of the small general purpose sound mixers like the Mackies or Soundcraft Spirit series are very useful when your main need is to actually mix several microphones together for, say, a panel discussion type of program, or when you're recording live music performances. They usually have more extensive equalization and signal routing functions than the mixers designed exclusively for video production, but as Steve says, you won't find battery power options on these boards -- they are AC only. They are also useful back in the studio for doing voiceover work. Most video production mixers like the Sound Devices or Shure FP's aren't really used to "mix" that much, but rather provide very high quality preamps, good headphone monitoring functions, and a convenient and ergonomically correct control set for the sound operator working in a fast-paced environment. In other words, they're designed for a different set of applications and priorities.
I have a medium-sized but still quite portable Soundcraft board with eight mic inputs and two additional stereo feeds for tapes, CDs and such that has been very useful when I have real mixing to do on location. (The Mackie 1604 and their higher-end Onyx boards are also quite popular for this kind of thing.) I also have three little Soundcraft Folio Notepad mixers -- tiny and cheap things indeed, but very handy for small jobs and as a kind of general-purpose interface between my video equipment and the various sound systems I find when shooting in the wild. On the other hand, it's my SD302 that travels in my bag all the time for doing quality sound-for-video with the more typical rigs of one or two handhelds, shotguns, hypers and lavaliers that I use on narrative, commercial and documentary shoots.
I do try to stick with mixers designed for recording applications rather than the ones optimized toward musicians' use for live sound reinforcement. Those are the ones that often have lots of (relatively low-fi) delay and reverb effects and other doodads that you wouldn't want contaminating a quality recording. They also tend to have lesser mic preamps since pristine noise levels aren't as highly valued in live performance situations.
you see mixers a lot in full carts. It's a fast, easy, more user friendly way to mix in my opinion. You need to have a setup that utilizes it though, and it's no good for the ol' run and gun.
My cart: focusrite preamps digital out to digi003 for recording isos
analog outs to mv802 mixer
802 Mixer
Main out L to digi003 to record mix
Main out R to camera
Aux 1 to wireless (usually to boom op)
Aux 2 to camera (second option)
An SB3 generates my timecode and acts as a master clock for the focusrites and 003 toslink connection
It's a metacorder setup that doubles as a mobile PT rig. (hence the 003)
There is also a lot more in there but most of it I try to keep kinda secret. I have a lot of neat tricks in there including wireless remote control via an android tablet.