No.
You assume that an increase in hoarding must come from an increase in savings but my sequence of events assumes precisely the opposite. You see the "spend vs save" but you don't subdivide savings into "hoard vs lend".
And you don't see anything other than lending as productive savings. This is far from the case.
No, I don't see hoarding as productive savings. Can you explain me how it works?
I doubt it, but one productive example of interest free savings is as a form of insurance. For example, certain religious groups believe that they are commanded to
prepare for foreseeable events. The bible mentions famines, fires, etc. So these denominations might expect each family to "stock up" on consumables, or an entire church might stock a food bank. Cash can be
horded for similar effect. Not for the goal of investment at all, but "saving for a rainy day". This kind of saving is self-insurance against whatever future losses that might present themselves. Individuals do this, so do corporations. Both do it using many methods; individuals could do it by stuffing cash into a mattress, buying gold or silver rounds, or a large pantry stocked with non-perishable. Corporations could do it by warehousing materials needed for manufacturing inputs, such as steel or plastics; by speculating on the futures markets for same, or by
hoarding cash, quite literally in a 'cash-on-hand' account; or by buying gold. Governments do the same thing, usually by hording resources such as oil.
Funds used in this manner are sometimes put into low risk investment methods, such as a money market account, but the general idea is that those funds should not be put at risk of loss, because they are insurance not investments.