I've done fmq with both types of bobbins with no problems, and always lower the feed dogs and use a quilting or hopping foot. Without the foot I find the needle and thread tends to pull the fabric up with them, even if it's hooped tightly. I believe eyelashing happens when the bobbin jerks instead of turning steadily, maybe from moving the fabric to fast or too far, also I have tried cutting a small circle of thin fabric or interfacing - with a hole in the centre if needed, place it into the casing before putting the bobbin in, and this seems to hold it firmer. If it's the top thread making the mess using a hopping foot should help solve it, besides it's slo easy to get a finger under the needle - ouch.
BTW I think all machines before early 90's had front or side loading bobbins, except treadles
Hope this helps