Can I use my Mac's speakers and AirPlay / Bluetooth at the same time?

Try using the midi setup application included in OS X it can let you group sound outputs together. You just need to create a new multi-output group with the + button.

The only way I can think of doing this is by using a 3rd party app "Airfoil". But if you use Airfoil you will have to use 2 different applications to play the music (eg. Spotify will stream to AirPlay while Chrome plays on Mac). I know Airfoil isn't the fix you're asking for but it's the closest solution you are likely to find.

https://rogueamoeba.com/airfoil/mac/

I don't have a rectangular Airplay button on Itunes, but if I click the circular airplay icon to the right of volume bar, I get multiple options to redirect the audio output, similar to the picture in the accepted answer (MacOS 10.14.6, ITunes 12.9.5.5).

However, as is mentioned in the comment to the accepted answer, my Airplay broadcast disconnects as soon as I try to change the audio output in ITunes.

I was, however, able to direct audio from VLC to my built-in speakers without affecting the Airplay output. I can listen to music on the computer while Airplay broadcasting TV to another room.