Easiest way to connect two non-thunderbolt monitors to my Macbook Pro 2017 and charge it at the same time?
Regarding your Satechi hub and the monitors:

"the monitor connected by Thunderbolt doesn't seem to be sending USB-A signals"

This is correct. You're not actually connecting the monitor by Thunderbolt, as the monitor does not support Thunderbolt at all. The monitor supports VGA, DisplayPort and HDMI, so you probably have a USB-C to DisplayPort cable or similar connecting the hub and the monitor. That cable will transfer a picture to the display, but it won't get the USB-A ports on the monitor working. To get the USB-A ports on the monitor working, you need to connect the upstream USB-B port on the monitor to one of the USB ports on your hub.

"More significantly, the USB-C port on the hub doesn't work for charging OR for display."

The USB-C port on the hub is not supposed to work for transferring a display signal. Only the HDMI video output port and the Thunderbolt 3 port on the hub have a display signal in them. This is also visible in the graphics in the link you have provided.

The USB-C port on the hub is also not supposed to work for charging. This is listed in the product description on your link: "Please note the second USB-C port is for data transfer use only." - i.e. no power delivery.

Alternatives

Instead of the Satechi hub product, you could look a other products that fit your requirements better:

For example the OWC Thunderbolt 3 dock takes up a single Thunderbolt 3 port on your laptop - leaving the other one free. It does so while providing a display signal for both monitors, charging your laptop and providing an addition 5 x USB-A ports, 1 x USB-C port, SD-card reader and Ethernet. It is priced at approx. 295$. 

Another example is the CalDigit Pro Dock that similarly takes up a single Thunderbolt 3 port on your computer while allowing you to connect two displays, charge the laptop and have access to 3 x USB-A ports, 1 x USB-C, Ethernet and SD card reader. It is priced at approx. 200$.

Multiple similar Thunderbolt 3 docks exists at various price points from various manufacturers.