I agree with your parrot, the more seeds the better.
In New Zealand we have a cucumber called the port Albert cucumber, its big, easy to grow, and has tons of seeds. Unfortunately I couldn't find it for sale outside of NZ (maybe under a different name?).
A slightly smaller international alternative, still with lots of big seeds (tho not quite as many), is the Crystal apple cucumber. Its also easy to grow.
We've grown both of them and will grow both again. They get more/bigger seeds when left to mature, but go sour if left for to long. I still love them when they are sour (your parrot might to) but everyone else thinks I'm crazy.
Bananas do and will loose their bottom leaves as they grow, so no surprises there.
As you probably know the black flaky stuff on the outside is normal sap, not mildew. As far as u can see there is only mildew present where the frond meats the trunk.
As there is only a small amount of mildew and no signs of rot I don't think the mildew caused the leaf falling off (rather just natural growth). If it gets more chronic mildew it may cause issues in the future. We grow our bannanas outside and powdery mildew between old fronds is a given, and not a concern, however indoor plants are a lot more prone to disease so it may be a worry.
P.s. its not powdery mildew (mildew is white/grey). Its probably just some kinda mold that settled in the crack left by the leaf falling off. I would only be concerned if it is present between young fronds or starts rotting the trunk.