Providing additional support for door hinges with no stud behind
Your idea isn't terrible -- it's really nice to have long screws through the hinges, but it still might be overkill.

Long screws through the main part of the jamb that pass close to tight shims and go into the stud will be sufficient. I'd do 4 per side. If you don't want to patch and paint a whole bunch, you might be able to take off the hinges and run a long screw on a slight angle that will catch the stud.

If it isn't an integrated stop as shown in your drawing, pull the stop off and put screws behind it.