Counterintuitive, sure, but less counterintuitively, it is also false.
> A rise in rates means more bond coupon and more bonds sold (new money)
No, it doesn’t mean more bonds sold.
(Purchasing bonds is lending; the idea that lending increases with both rate increases and rate decreases is…wrong. Borrowing, whether via banks or via bonds, is more common when it is cheaper because of low rates, and less common when it is expensive because of high rates.)
And exchange of equity instead of interest for money follows the same patterns, because those with capital will trade it for less valuable (in expected future value) equity when they’d make less money in its alternate use (lending), and demand more valuable equity for it when they’d make more lending. So, equity financing (which, despite the structural difference, also gets the money moving in the economy) is also more active with lower rates and less with higher rates.