Do these 2 x 4’s provide structural support

I think they're there for stabilizing the brown beams/joists. Typically, a loft is unused space with no floor. Between the joists, there is insulation supported by drywall or other material depending on the age of the house (as in pic 2 and 3).

Typically one would put in blocking between joists to stop horizontal flexing and torsion.

Since an attic is usually unused space, I think someone took a short cut and just screwed the 2x4s to the tops of the joist. (Significant labour savings!). Your flooring probably makes it unnecessary from a flexing point of view, but you still have a torsion problem (especially since you're now loading the joists).

When you go to make improvements, block the joists properly, but I wouldn't worry in the short term.

Since you are in zone 4, R-38 is a reasonable insulation level. That doesn't tell us how much an issue a lack of vapor barrier is. It is less of an issue in dry climates. What a barrier does for you is prevent moisture from condensing inside the insulation, significantly reducing it's insulating value. Ever notice how hypothermia victims have invariably gotten themselves wet? Unfortunately, properly installing one can be very difficult (read: expensive). As it happens, plywood acts somewhat as a barrier, depending on it's construction. Thus, you may need to just focus on gaps and openings.

Another possible make-do vapor barrier is a good heavy coating of latex paint on the ceiling. You may already have this! Truth be told, the best way to insulate would be to remove the plywood and blow in the needed insulation. This gets sealed around all the joists and other ins and outs nicely and makes effective use of what little insulation is already there. However, it may not be worth the effort.

You can still get a good insulation job by layering it on top as you suggest. The draw back is the dead space under the plywood effectively becomes part of your conditioned space, greatly diminishing the value of what's there. Even a plaster ceiling alone is worth something. And you've increased the top story volume you're heating 5-6%. Another issue with this approach is the plywood becomes your vapor barrier. What are the ramifications of moisture condensing on the under side? There's poor air circulation, so it will not dry easily. It could be the cause of a mold colony gaining a foothold, which can have serious health consequences. Or it may not be an issue if you're in a dry climate.

Whichever way you go, try to ensure what ever passes as a vapor barrier is a complete as possible. Ensure any recessed light fixtures are not insulated over, and that no insulation contacts the cans. It may seem more efficient to cover them, but you greatly increase the chance of them overheating and starting a fire. Insulation is little good if your house burns down! A truly efficient ceiling has no recessed light fixtures for this very reason. Note that there are supposedly fixtures rated to be covered, but I don't think I've ever seen one.