The title pretty much describes it. So, should I connect my older, weaker modem (which in its own can broadcast a wifi signals) to a new router that can broadcast much better through an Ethernet cable? If so, would the signals get better or would it stay the same?
If it has wifi it’s not a router it’s an access point, possibly a router with an access point built in
Modems MODulate and DEModulate.
That is correct, however what you said previously is false as a router is nothing but a firewall and directs network traffic. It has 0 responsibility in providing a wireless signal. hence why we have Wireless Access Points (WAP) so one can wirelessly access the network