[RPG] What happens to a person who is scattered to the astral plane when inside of a destroyed bag of holding

Yes, you can use astral projection from planes other than the Material Plane. The Manual of the Planes describes how spells that access transitive planes such as the Astral Plane function on other planes in the section "Moving Among Transitive Planes" (starting on page 45).

Characters generally must use spells or spell-like abilities to access a Transitive Plane. ... the astral projection spell takes you to the Astral Plane ... Such spells should function in any location coexistent with or coterminous to the plane.

It includes various examples of travel between transitive planes on page 46. These examples reinforce that "Material Plane" is a placeholder and that you can use astral projection from anywhere that is connected to the astral plane. Your body will be left behind on the plane you project from.

From the Ethereal Plane to the Astral Plane: You can move to the Astral Plane from almost everywhere on the Ethereal Plane, because the Astral Plane connects to everywhere on the Ethereal Plane. ... If you're using an astral projection spell, your physical body remains in the Ethereal Plane ...

From the Plane of Shadow to the Astral Plane: From the Plane of Shadow, you can use the astral projection spell, leaving your body behind on that plane ...

D&D 5e made significant and substantial changes to nearly every major spell in order to accomplish its systematic rebalancing of the magic system. In all previous editions of D&D (except 4th, which completely replaced the entire subsystem), Astral Projection allowed you to visit other planes in such a manner that your death on the visited plane would leave no serious lasting harm. D&D 5e did, in fact, change that (at least in the spell description), but the author of the section on planar travel in the DMG is clearly writing with an understanding of the previous way the spell worked. It's quite probable that the person writing this section was a different author, and possible they had no idea the spell had been so significantly altered. In any case, I think the update in the PHB is consistent with the updates to magic overall, and should be preferred over the text in the DMG, which makes incorrect reference to the spell as a related topic.