You are confusing or lumping network speed with internet speed, which are rarely the same.

For example: My home office has multiple computers connected to each other via a 1 Gbps wired network router. My network also includes a 300 Mbps (max) wireless access point to connect WiFi devices. The entire network is also connected to the internet via a DSL modem with 20 Mbps download and 750 Kbps upload. If I was going to extend my network I'd want to come as close to the highest existing connection speed as possible (1 Gbps in my example.)

Some network devices include wired network, wifi network, and internet modem in one device. But the varying speeds and concepts are still the same.

If you ONLY connect the extended devices to the internet it's mostly a moot point. I don't think they make extenders with network speeds as low as most wireless, cable, or DSL internet speeds.