In my experience, an axle mount is most stable compared to a seatpost mount.
With the seatpost clamp, the differences are most evident when braking and when turning. The contact point is a hefty "shove" from the side/rear, and because its so much higher than your wheels it has enough leverage to be noticed.
Imagine, turning at any speed and for some reason the trailer hangs up on a bump or pothole for an instant. Your saddle ends up being "pulled" to the inside of the turn.
Imagine when braking, your free-running trailer is shoving you up the seatpost and fighting the brakes. If you're hard-stopping so 90+% of the braking is on your front wheel, then the back wheel can be lifted clear off the ground and get pivoted sideways by the trailer.
When towing, rear wheel braking increases in importance from "almost useless" to "quite a good idea"
Note I've never tried a BOB-type connector where the pivot is behind the rear wheel. I imagine these feel like axle mount trailers.
A chainstay clamp might be convenient but mine almost fell off several times, so upgraded to a towball and hitch for safety.