Your requirements (routing + NAT at 1 GBit/s WAN speed) and budget make this kind of difficult. Yes, the figures provided by @dlakelan and others suggest the RPi4 to cope with that kind of routing throughput, but its wireless capabilities are non-existent (don't count its SDIO based WLAN module. It can only handle a single channel - and at draft-n (read slow) wireless speeds/ low range (due to the inadequate antenna)), so you will have to include a decent 50-80 USD AP (router in AP configuration) into your calculation (ipq40xx comes to mind, maybe mt7621 or ar71xx if you're willing to makes sacrifices for lower device prices).
You might still end up with a RPi4 to deal with your requirements, but in order to service your WLAN clients, you will have to extend your budget for the cost of APs as well (which gets you closer to 150 USD). In theory, mvebu devices like the Linksys WRT3200ACM/ WRT32x would fit this niche pretty well (it's supposed to handle routing at 1 GBit/s linespeed), but sadly its wireless driver development has been discontinued (due to Marvell selling their wireless division to NXP), despite it still being pretty buggy and uncooperative with common IoT/ smarthome devices and without functional WPA3 support.