The blocking is only providing lateral support to the joists, so I don't see the need or benefit of attaching it to the studs. Install 2x4 (full depth) blocking on top of your existing ledger and tightly between the joists. You can end nail one side of the blocking through the joist, but you'll need to toe nail the other side. Then install the 2x4 flanges to the bottom of your joists as described above. Use plenty of adhesive and fasteners.
When building a storage loft, it is highly encouraged that you make it as durable as possible because you do not want your belongings which are stored inside to be tumbling down in just a matter of months. You want them to stay safe up there for as long as possible. Thus, building it like as though you are protecting yourself from a nuclear attack is a good sign. Even though it might not be the most practical solution yet, but it is the safest. So, you got to know your priority really.