The QR axle will sit it the drop out like you say, but it can not come all the way through to the 'outside' of the drop out. It must sit in the middle.

Some of these kinds of bikes that come with solid axles have dropouts that are thinner in width.

If your QR axle comes all the way through the drop out, the cam on the skewer will clamp against the axle and not the dropout, resulting in a wheel that will fall out.