3DS doesn't exactly have a lot of output ports, just a headphone port and a charging port. So I'm not sure how a capture card would even work on a 3ds. Regardless, if it's doable, it would be much better.
As far as "best camera for $175" in general goes, I'd say an old used dslr+basic lens could easily be had for that price. But that probably wouldn't fit your needs as such old dslrs usually don't have video.
EDIT: ok i know like 0 about mirrorless models but
this body (used) with
this lens (used) looks fine. Would only be 720p video tho.
Don't bother buying a point and shoot like scythe suggested. Nobody buys low-end point and shoots anymore because cell phone cameras are pretty much just as good; if that's the option you want to go, just stick with your phone. You'll get much better results buying (relatively) higher end used equipment like what I suggested
