USB labels are confusing these days, and your post leaves me with questions.
First, ALL USB 3 systems now are USB 3.2, with a Genx appended. The Gen1 version can deliver up to 5 Gb/s max data transfer rate in the communications subsystem. The device may NOT reach that speed because its components can't move that fast (e.g. a really fast mechanical hard drive can only get to about 2.5 Gb/s), but the communication subsystem is deliberately designed NOT to be the speed limiting factor. The Gen2 systems can do 10 Gb/s max data, and the Gen2x2 systems can get to 20 Gb/s. For cables and CONNECTORS, the old Type A for USB2 connectors now have BLACK plastic inserts and the new versions of these with more contacts in them have BLUE inserts - they can fit into older USB2 sockets, too. These Type A connectors are fine for Gen1 and MAY be used for Gen2 (speed not guaranteed), but it is really recommended that you use only the newer smaller Type C connector for Gen2, and you MUST use Type C for Gen2x2. On a Motherboard, the USB 3.2 port headers for these three Gen's are all different because of the controllers. Mobos use the SAME header types for Gen1 and Gen2, and each of these headers actually contains TWO USB ports so its cable normally connects to two external sockets. The sockets can be older Type A for any Gen1 port. For Gen 2 ports they should be the newer Type C, but many mobos use one Type C and one Type A for these, just to make it easier for users with older equipment to plug in. On a mobo the Gen2x2 header is very different - called a Type E header which contains only ONE port at this high speed max of 20 Gb/s, and it MUST be connected to an external Type C socket.
Your post uses the term Type C in the headline, but "Type 3" regarding mobo headers. The mobo MAY have labels that say USB 3.1, instead of 3.2, and they may not use the Genx label system. But at least they should tell you max speeds of 5 (Gen1) or 10 (Gen2) Gb/s. It is the mobo HEADERS and their CONTROLLERS that determine what speeds your external sockets can do.
So, if ALL your mobo headers are USB3.2 Gen1 (max 5 Gb/s) ports, then you need only the USB3 version (blue inserts) of Type A sockets and cables - these are fully able to work at that port speed. IF you need to connect to a Type C connector on some cable, you can get simple adapters to convert a Type A socket into a Type C socket.
IF your mobo has at least one header that does operate as Gen 2 (10 Gb/s max), then you should be getting some way to feed its two port signals to one or two external Type C sockets. Although you can use Type A sockets you may or may not get that higher speed.
IF your mobo has only Gen1 headers but you want to ADD a pair of Gen2 ports, then you need to start looking for a PCIe card to add that type of Controller chip and connection sockets to your system.
Unless you have a Type E mobo header built in that offers a Gen 2x2 port, I have not yet seen any PCIe card that can add that to a mobo.
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