No. Over-current trips do not flip the breaker to the off position. Instead, it trips internally and the handle moves ever-so-slightly and becomes “rattle-y” for lack of a more precise term.

If you have a breaker in your hand (now or at a home center), you can slap it briskly against your thigh and it will “trip” allowing you to see how the handle will look when tripped. To a cursory glance, it will appear on (but you will notice the handle can be moved about 1/8" more than normal).

That small movement is not enough to induce a trip or disconnect on the handle-tied breaker. (The requirement is only "Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates." That's to allow for safe servicing.)