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?
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
But he was responding to the op who referred to his modem as having weak wifi, therefore correctly stating he most likely had a modem,router,ap combo aka a gateway.