Thank you for your answers, Jason and Alberto.
While it's true that the hypercard is a PCIe 3.0 device, I did not see the reason yet why it would not operate in a PCIe 2.0 slot with lower throughput. And I am not sure but - is Intel VROC required to use all four M.2 slots?
Anyways, I am happy to report that I "resolved" the issue partially by getting rid of the GPU that occupies two slots, replacing it by one that occupies just 1 slot. Then I installed the USB3 controller card next to it, resulting in the following setup:
1. (MIO): empty 2. GPU 3. USB3 controller card 4. Hypercard 5. Empty
This way, I have graphics, USB 3.0 and two M.2 slots working. In addition, I noticed that the system (without usb3) takes around 7-8W less in idle than with the previous gpu installed. That's a bit surprising as the old gpu was advertised to use only ~5.5W total while idle (but I wondered while it gets so hot, I thought this was due to it being installed so close to CPU0's huge heatsink). As I don't need that much GPU power anyways, the swap had at least a nice (unexpected) side-effect for me.
However, the fact that I still can use only two M.2 cards is somewhat disappointing. Since there are also other vendors of comparable PCIe quad-M.2 cards, I welcome any reports (both positive and negative) from people who tried different m.2 quad card models in a KGPE-D16.
Cheers, Daniel