I use the little swinging flap, but also a springy strip of stainless steel attached to the rudder head and which braces itself against the gudgeon unless pushed in. More convenient than a cotter pin, but also more positive than the little swinging flap. . .
It takes very little force (given the long lever arm represented by the mast) to balance a boat sideways. For the same reason, very little flotation is needed to resist or slow turtling. You can estimate the righting moment by taking the volume of your mast profile at each distance from the bottom and multiplying it by said distanced and then summing the result. I haven't done the exercise but am willing to believe that the mast, if water were to be kept out reliably, would make a good part of the required flotation.
Your suspicion that you have 60 year old styrofoam that's no longer "floaty" is most likely spot on. Pool noodles are the modern choice for replacement. (Sealing the entire mast would work, but would need to be perfect!).
Next, having the gunwhales awash is indeed a problem, even for modern boats. A friend had one of those ubiquitous Walker Bay dinghies (with a sailing rig) and heeled it too far and it filled. Stayed afloat, but he was stuck because he couldn't bail it. The mid-sixties DS1 has three flotation chambers. One in the bow, and one under each seat. These chambers are filled with aging foam that may well be waterlogged, therefore non-functional.
If restored to their intended level of functionality by making sure the chambers are dry and sealed with non-waterlogged foam (pool noodles), they will generate positive flotation. Should be pretty straightforward to work out, from the volume, how much flotation in relation to the approximately 5-600 lbs of hull and rigging. (I haven't run those numbers, but perhaps someone will).
Now, I have a hunch that with flotation chambers working correctly, you could have just opened the transom drain in the back and the boat would eventually float higher in the water, up to the point where the combined weight of that part of hull and rigging above the water line equals the flotation. The reason you didn't see that in your case is that raising the boat would mean raising the trapped water. (If you open the transom drain, it can escape, until everything is equalized).
To finish bailing, you'd need either a self bailer or a 5 gal bucket. To use a self-bailer, the boat would need to move, of course. For a bucket, 5 gals is about the best size for bulk bailing, but you'll need something else for once the puddle gets shallow. So far the theory. I wish someone would run some of those numbers to see whether this could even work.
When it comes to practice, I've subscribed to a strict "no capsize" policy