The problem described in the video is, that nodes in the two different IP subnets have to be aware of the routing. That is not transparent for the main router (that one with the real internet access) makes it difficult to configure. Also DNS-requests in the internal network have to be answered by somebody.
Things are a lot easier with a NAT-router that acts in the same way as all common WiFi DSL or cable home routers do. In order to turn the ESP into a NAT router some additional code in the TCP/IP stack is required, namely the NAT translation table.
You might want to try this: https://github.com/martin-ger/esp_wifi_repeater. It turns the ESP into a full functional NAT-Router. Performance is quite reasonable - youtube clips e.g. are possible.