As a broad statement, this is a terrible idea. Why? Your truss system is designed specifically to hold up the roof and, to a lesser degree, tie your walls together.
2x4s spanning 28' with breaks are clearly not up to the task of holding up plywood and sleepers (parenthetically, not sure why sleepers seem like a good idea, but I digress), much less whatever stuff you (and future occupants) load up there. It simply isn't in the design.
What you'd need is something resembling a floor system... sturdy beams running across the 28' dimension with reasonably sturdy lumber running perpendicular to the trusses. (That beam at the front, by the way, is going to interfere with your garage door/opener.) Or some engineered joists running parallel to the trusses, which might or might not be able to simply bear on the top plate. On top of that, we need to ensure that the load on the joists/beams is being carried appropriately to the floor through posts, which then need to be carried adequately on a footing.
So, ask yourself: is all this worth the trouble? You'd lose either headroom in the garage or headroom in the "attic" because of a need for decently sturdy framing lumber. Cost of time and materials isn't trivial. You might need to revisit your garage door opener mechanism. Ensuring adequate bearing for joists or beams isn't going to be simple. You're probably digging new footings in your garage floor.
Storage sheds or offsite storage units start to look pretty good at this point.
So, capsule summary: this can be done. You need actual engineering expertise to do it appropriately. It'll cost a lot.
The alternative is to just chuck some plywood up there; undermine the structure of your garage; then call a contractor in 5 years to fix it all at great expense.
Disclaimer: I'm a professional carpenter, and I'd rather not have to charge people money to fix things like this.
I would sister several of the joists, then add solid blocking in each span. The blocking might not be critical, but the ceiling is now open so why not? Solid blocking does not need strapping across the bottom.
Look closely where the joists meet the load-bearing wall at each end. A 2x6 ledger might give extra assurance. Look at the rest of the floor structure for any splits, sags, movement, etc.
Best Answer
It should make you leery. Playing around with a wood beam calculator I had to trim loads down to 7 lbs per square foot to get any of the solutions to come up 2x6 for 20 foot span and 16" spacing. That's not even light storage (by my definition.) It's not even a residential attic with no storage by more official definitions (10 PSF) nor an attic with storage by those (30 PSF)
Either leave it alone or hire a civil/structural engineer to develop a plan where you'd either refit the roof structure (effectively turning a rafter and joist arrangement into trusses, if that's practical, by adding web members) or place a properly sized and supported beam down the middle to cut the span in half. In short, if you want storage up there, it will cost a bit more than a set of stairs.