The OP seems to think his USBa connectors cannot connect to USBc ports.
No, OP is asking whether he'd get better performance plugging his devices into a USB-C hub like the one he linked to rather than his existing USB-A 3.0 hub (Spoiler: no, but...).
My question is if I connect any of my USB 3.0 peripherals to any of the 3 USB 3.0 ports on that device, which then plugs into one of the USB-C ports on the Mini, will I see a dramatic increase in speed for that connected device?
No. Certainly not that device. Most such "USB-C" hubs are exactly equivalent to a USB 3.0 hub plugged into an old-school USB 3A port (well, they can add charging and DisplayPort/HDMI but that's not important right now).
Also, if I have 3 USB 3.0 devices connected to those 3 uSB 3.0 ports, will there be any speed degradation for any of those USB devices?
Yes - but the same goes for your existing USB hub. All USB 3.0 hubs (or USB 3.1 gen 1, which is essentially the same thing) - whether they use a "USB 3 A" or "USB-C" connection to the computer - share a
single 5Gbps connection with the computer between however many devices you plug in - so you can't have 3 devices all getting 5Gbps at a time.
If you want to optimise your existing set-up then prioritise your devices and connect the ones that need high speed or low latency (external SSDs, audio interfaces, high-speed SD readers for video/photography etc.) directly to the Mac, either to the two USB-3 ports or, using the USB-A to USB-C adapters that people have linked to, to the any of the USB-C ports that you're not using for displays. That way they're all getting full USB 3 speed (but nothing faster). Then use your existing hub for the 'don't care' devices like mice, keyboards, printers, thumb-drives, regular card readers etc. Sure, you are using up your USB-C/TB3 ports for boring old USB 3, but its still more efficient than connecting
everything to a single USB 3 socket. Whether you'll see a noticable difference with your devices is highly questionable, but you'll sleep better at night and some devices have "latency" issues or compatibility problems with hubs.
(I've spent a lot of time whining about USB-C ports on laptops where you have to cart around otherwise unnecessary hubs or adapters, but even I don't really see the problem on the desktop).
Now, if you want to use the "extra speed" of those USB-C TB3 ports, then you want either:
A Thunderbolt 3 hub - like the CalDigit TS3+ - although that gives you lots of stuff like ethernet, DisplayPort and laptop charging that you don't want and costs a lot.
A Thunderbolt 3 to USB 3.1 adapter/hub like this:
https://www.startech.com/Cards-Adapters/USB-3.0/Hubs/thunderbolt-3-usb-3-1-hub~TB33A1C - looks like a regular USB-C hub but isn't (...but probably the Caldigit is still better value).
Maybe A "real" USB 3.1
gen 2 hub like this:
https://www.lindy.co.uk/usb-c4/4-port-usb-3-1-gen-2-type-c-hub-with-power-delivery-p11211 - big "beware" on this, but a man in a pub told me that true USB 3.1g2 hubs
can actually share the extra bandwidth of the faster 10Gbps USB gen 2 between multiple 5Gbps 3.1g1 devices (...i.e. each device can't have more than 5Gbps, but the total over all devices can be more than 5Gbps). Maybe others here can confirm it (I'm taking it with a pinch of salt at the moment).
Finally, remember rule 1: if it ain't broke, don't fix it. Are you having speed/latency problems? I wouldn't expect any night-and-day improvements from the above, and there's lots of factors that can affect the speed of your peripherals.