New modem; router can't connect to internet, disconnects during setup; ethernet fine

> We had to replace our modem [...]

   What was it?  Maker?  Model number?  Was it actually a simple "modem"
or a modem+router?  Did you connect multiple client devices to it?

> [...] New modem [...]

   What is it?  Maker?  Model number?

> [...] only had one ethernet port, [...]

   So, probably a simple modem.

> [...] so we bought the GS108 ethernet splitter. [...]

   Unwise.  A modem plus a network switch (not called a "splitter") does
not replace a modem+router.

> [...] This was connected to our Nighthawk X6 R8000 router and wifi was
> setup just fine.

   What, exactly, was connected to what, exactly?  (Hint: If a device
has different types of ports, then "connected to device" is not enough
detail.)

> However, [...]

   All as I'd expect.  Your ISP is providing only one external/public IP
address, so you can't connect multiple devices directly to your modem.
Adding a network switch doesn't help.  Adding a NAT router does.  You
need to connect the router (WAN/Internet port) to the modem (LAN port).
As shown in all the product documentation.

   You can connect (any one port on) the switch to a LAN port on the
router.

> Here's what I've tried so far: [...]

   All pointless.  Lipstick on pig.

   Visit http://netgear.com/support , put in your model number, and look
for Documentation.  Get the User Manual (at least).  Read.

   If your R8000 was previously connected to a modem+router, then it
might have been configured as a wireless access point, rather than as a
full-function router.  If so, and that other router is now gone, then
you would need to put the R8000 back into full-function router mode.  A
settings (misnomer: "Factory") reset and reconfiguration might be the
best way to do that.