The flaw of the free-sprung balance

Who says this?

IMO the primary advantage of the free spring balance is for isochronism. Regulator pins act on the balance spring to artificially limit it's effective length. At a good balance amplitude, the balance spring will bounce off each side of the regulating pins, but when the amplitude drops enough that the spring is no longer contacting those pins, the effective length of the balance spring suddenly changes. This can lead to very sharp changes in timing.

This is the stud carrier. On modern watches, this can be moved, which is a convenience for the watchmaker. On older watches the stud carrier is fixed, so it cannot be moved - on those watches you have to rotate the balance spring collet on the balance to adjust the beat error, a process that introduces risk into the servicing component that doesn't need to be there.

In my experience is pretty rare that either the regulator of stud carrier is moved through a shock. If the shock is big enough to move these, you likely have other problems.

Different makers use different designs. So adjustable mass balances come with screws on the outside of the balance rim, on the inside of the rim, some are screwed into the rim in tapped holes, some aren't screws at all but are eccentric weights that can be rotated on a small post.

The difficulty with all these systems is adjusting two opposite weights perfectly evenly. In the hands of an amateur or a watchmaker with lesser skills, a poise error is very likely to be introduced when the time is adjusted. Aside from that, having to adjust a necessarily quite tight screw or weight on a balance in situ, significantly increases the chances of damage to the balance spring or staff.

Note that these screws can move, in particular on older watches if the screw has been moved around a lot. The thread fits on these are intentionally tight to prevent this, but repeated use can cause them to be less secure.