How do you implement wifi with a front-end chip?
Short answer: you combine it with a backend :-)

The MAX2831 you mention is a transceiver and that is all it does. As the manufacturer says, its a "frontend". What you miss is the "backend". Guess which part is even more complicated ? ;-)

The frontend contains only the radios needed to send (TX) and receive (RX) ! It does NOT generate the WiFi signal, I mean, in principle (ignoring some details) you could also use it to send and receive Bluetooth or Zigbee signals (which will need to be generated by a backend IC).

In other words, what is also needed is the modem. The modem is the backend and it takes care of the Wifi protocol, what frequency must be used (it controls the frontend besides sending and receiving the data) sets the bitrate etc, etc...

Indeed as pcj50 suggests, you're far better of with a complete module. Also, this will be cheaper as there are special components required to build a complete working Wifi module and these components might not be that easy to acquire and/or very expensive (as you would probably buy 1 instead of 1000).

Another thing is that on the software side the ESP8266 will be far easier, anything else (not a module I mean) and you might have to write your own drivers to talk to the backend. Ouch.