GS305v3 will not connect

With the unmanaged switch showing link LEDs and some activity for each port, here again the problem is somewhere in the "fine print" of what a Cable Modem does, respectively what services the ISP does enable. In general, a cable modem on a consumer (and many basic small business) configuration is limited to one MAC address - typically to connect a NAT router. Slightly uncommon in your use case, it's odd that the unknown modem from the unknown ISP does assign just one private LAN (RFC1918) address. More common are public IPv4 addresses or worst case carrier grade NAT addresses from reserved public IP address spaces.

The reason the second device does get a 169.254.239.32 address, which is from the so-called ZeroConfig IP range - in fact a fall-back if there is no DHCP assigning a workable address to your devices connected, and the cable modem won't provide an additional "workable" address.

Fist step is to talk to your unknown ISP, the provider of your Internet connection and this unknown CPE (Carrier Premise Equipment) of what you are designating as a Cable Modem.

A possible answer is that the wall outlet providing this Internet connection is coming from a building- or apartment-operator, and it's intended to connect your own NAT router.

To learn and understand more, please also share the name of your service provider and the location once you have figured out what is going on.

According your description with the link LED and activity on all ports, this is not a problem or limitation with the Netgear GS305v5.