Your first idea is pretty common. There's usually a very coarse "localizer" scan at the beginning of a session, which is used to set the field of view for subsequent runs. The whole body scans are (at least in theory) meant to find tiny things that are asymptotic though, so I'm not sure that searching through (say) 10mm slabs will help much.

Open, upright scanners do exist, but they're lousy. The goal of the (big) magnet is to produce an incredibly strong, uniform magnetic field. Due to physics, this is much, much easier to do with a torus-shaped magnet than a 'U'shaped one. Even so, there's one point ('isocenter') where the magnetic field is maximally flat and the quality will be best. The gurney moves to point the region of interest (e.g., your head) right to the isocenter. That's why the tech often uses a little light or laser to find your position, rather than just asking you to scooch. Motion is also, as you alluded, a huge problem.

I hate to be discouraging, but I am excited to see people actually thinking about MRI on HN!.