If I understand you correctly you are using ethernet to connect to the modem since you only need a router if you want to use wifi. So, so long as you are happy connecting by ethernet then you don't need a router.
Not quite true.
A router (ignoring the wireless part for the moment...more on that in a second) serves two basic functions.
The primary function of a NAT (network address translation) router (which is what typical consumer level routers are) is to route an external IP address to various computers on the local network with local IP address, which then basically allows multiple computers (and other Internet connection devices in today's world) to connect to the Internet through one external IP address.
A secondary function then is to get a firewall effect. As I understand it, NAT routers are not true hardware firewalls in the traditional sense of a hardware firewall, but they do effectively behave as a type of firewall. They in effect "close" ports to external initiated Internet traffic, while still allow internally initiated traffic (i.e. if you go to a website, you are initiating the connection and then "two-way" traffic can ensue, but an outside source can not initiate a connection unless you tweak some of the router settings such a port forwarding or using a DMZ zone).
Now, in today's world, a typical "router" is actually a combination of several devices: the router function itself (this allows multiple Internet connected devices to access the Internet...i.e. it "routes" the connection), a network switch (gives the router multiple ethernet ports...original routers came with only one ethernet port and you provided an external switch), and a wireless access point (when WiFi was first released, you would get a separate wireless access point, but then routers started to come with them included as they are now...you can still get just "pure" wireless access points). Many routers also now come with printer servers (i.e. allow you to connect a USB printer to the router so that you can print over the network) and network attached storage functions (i.e. allow you to connect a USB hard drive to the router to allow you to access that hard drive over the network). You can also get routers that have "true" hardware firewalls included. And of course, you can now typically get modems (cable or DSL or Fiber) that include routers (with WiFi and multiple ethernet ports), which are typically then called "gateways" by ISPs.
As you noted in your additional post, many people now get a "residential gateway" (aka modem/router) from their ISP when they get Internet service. If one does, then there is no technical need to get a "third party" router, but some people (such as myself) due because the gateways from ISPs usually have weaknesses (in my case, the WiFi from the ISP provided gateway sucks) or "third party" routers might have more features.
In the case of the original poster, it sounds like they might just have a modem with no router function built in. If that is true AND the original poster ONLY wants/need to connect a single computer and is fine doing that by way of an ethernet cable, then the only potential reason to get a router would be the firewall function it can provide. Today's operating systems, however, do tend to provide rather good built-in software firewalls, so a router might not be needed...but it would somewhat depend on which version of the OS the person is running. Of course, there are also "third party" software firewalls for the Mac (as well as for Windows). I personally use NetBarrier from Intego on my Macs as it is easier to configure than the built-in macOS one (although the built-in one is more than powerful enough if using one of the newer maOS versions). And in my case, I also then have my router acting as a firewall too.