This stuff should generally "just work". I have a PCIe extension cable which is simply 30cm or so of ribbon cable with PCIe x1 connectors on both ends and it works. There is a lot of similar products based on USB 3.0 cables and AFAIK they work too. They are typically used to connect lots of GPUs to a single PC for crypto minining.
The rest of your product is a PCIe to PCI bridge. These also should work without problems and without extra drivers. Such bridges are sometimes added to PCI cards to cheaply turn them into PCIe products and most customers don't even notice.
It would be my guess that most if not all of those bridge chips implement one PCI bus, so you only have 133MB/s total BW available for all cards if there are multiple slots.
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I have one of those and it works. But it only fits low profile PCI cards. A full width card could perhaps be inserted, but there would be no obvious way to mount it securely and no access to connectors near the edge (if any).
edit
One thing I'm not 100% sure is if those PCIe/PCI adapters provide -12V power to the PCI card. There are no obvious switching regulators visible on the blue one above, but one may still be hidden somewhere - between the slots or on the bottom side. Not all cards use -12V, but some do.