By definition, anything allowed by an emergency authorization isn't fully approved. Whether you define testing that won't be completed for another 2 years as "skipped" or just "not complete" is irrelevant -- it's still an untested experimental drug or it wouldn't require an emergency authorization.

The one thing skipped is the progressive nature of the testing. Typically, phase 1 justifies phase 2 testing, and phase 2 justifies phase 3 testing. Under project Warp speed, this progression was largely bypassed so that different phases of testing could overlap.

Phase 3 testing wasn't even remotely close to being done by the time of the emergency use authorization. Phase 1 and 2 testing was only just done by the time of the emergency use authorization. The testing wasn't done yet, and wouldn't be complete for years. It wouldn't be done until after the pandemic was declared over. It doesn't matter if it was skipped or not, it wasn't completed.

The fact that two of the four major vaccines ended up being pulled off the market because they were unsafe should be definitive proof of my point. The testing wasn't done, and so these potential risks weren't caught during testing, and so dangerous products were put out to market, and after it was discovered those products had their emergency use authorization rescinded.