I think there is some confusion over which network we are talking about. You have your LAN (local area network) - that is, everything on your side of the "gateway" device, typically the modem. The other network is the WAN (wide area network) or specifically in this case, the Internet.
The performance of your local network is based on the weakest link in your local network. That does NOT include the modem (see note below). The performance of your Internet is also based on the weakest link but in this case that could be your modem, or it could be (and likely is) the performance of the service your ISP is providing.
Note: It is important to understand the differences between the various network devices.
1. A modem (modulator/demodulator) is used as a "gateway device" to provide access to the ISP's network and the Internet.
2. A router has just one “wired” input and one “wired” output and is used to connect (or isolate) two networks. In a home network a router connects "your network" (the router and everything on your side of the router) to the ISP's network (and the Internet) through the gateway device/modem.
3. A Ethernet switch (also called a switching hub) is used to connect via an Ethernet cable, multiple networked devices on your network to the one input port on your side of the router.
4. A WAP is used to provide wireless (Wifi) access to your network. The WAP actually connects internally to a 5th Ethernet port of the switch. An extender is a type of WAP.
Technically speaking, there is no such thing as a "wireless router". That is simply a "marketing" term for a "3-way" "integrated" device consisting of a router, a Ethernet switch (typically 4-port), and a WAP. These are three
discrete network devices that just happen to share the same box, same PCB (printed circuit board) and same power supply. Note the 4-port switch connects internally to the router's one input. And the WAP actually connects internally to a 5th Ethernet port of the switch.
In recent years manufacturers have also been integrating a modem with "wireless routers" for a 4-way integrated device. Four
discrete network devices in one box. These are "marketed" under different terms like "residential gateway" devices, "wireless modems" or " wireless modem/routers".
In some cases, they even have 5-way devices that integrate Internet phone (voice over IP or VoIP) too. Five discrete devices, in one box.
I mention these differences because terms are often intermixed incorrectly. People will say modem when it is really the integrated router they are talking about. Or they say router when it is really the integrated WAP. So we really need to be talking about the "function" as well as the device, and not get them mixed up.