How to turn off AirPlay on Apple Watch

First thing I'd check is whether the firewall is enabled on your Macbook and blocking incoming packets. If the firewall is on, try turning it off temporarily for a few minutes to see if the AirPlay icon appears in the menubar.

Next thing to check is whether the devices are on the same subnet. You can find the IP address and subnet mask of your Apple TV in Settings -> General -> Network. And the IP address and subnet mask of your Macbook in System Preferences -> Network -> Wi-Fi -> Advanced... -> TCP/IP. Ideally what you should expect to see is the same subnet mask on both devices (likely something like 255.255.255.0) and IP addresses that are similar except for the number after the last dot (something like 192.168.0.125 and 192.168.0.138).

Those are the two things I'd check first. Check those and let us know how you get on.

Apple have a guide to trouble-shooting AirPlay problems for Macs. There are a few other things that might help too: About AirPlay Mirroring in OS X

The short answer is, that you are doing the right thing, the Apple TV should just wake up when it gets an AirPlay stream coming at it.

My Apple TV (720p) one is having similar issues, you have to wake it up to stream to it, and it has other issues.. I would suggest a factory restore which is what I am going to do.

It will loose all your preferences, but should fix the issue. Worst case you could also look at reinstalling the firmware using a Micro USB cable and iTunes on your PC or Mac.

Instructions on Apple.com for both methods