Is there a way to make playback happen between the L and R locators? That would be very useful.
(I know you can loop playback between the L and R locators, but I’m talking about a one-off play. And I know you can use the selection tool to do something similar, but I don’t want to have to do that, I just want to be able to press a button and get the playback to happen between the locators.)
Really? Selecting a Part that encompasses the desired Range, then hitting Alt+Space (by default) is not convenient enough (although of course “Play Locator Range” would indeed be a nice simple addition to the commands 
)?
There are other ways (e.g. using the Arranger Track), but, frankly, more work to set up than just described .
To complement your comment, isn´t there a command to Select Part Within Locators that OP could use first? That should take care of his needs.
I think that Command is named “Select in Loop”, but it would only be of use in this case if there were already a Part that was exactly between the Locators… it would also select any Part even if the Part started before the Left Locator, or finished after the Right Locator, or indeed if there were space between the Left Locator and the start of the Part, or between the end of the Part and the Right Locator.
The Alt+Space command would then play the range of the selected Part(s), not necessarily from Left Locator to Right Locator.
However, there is something …
(although not sure if this would in fact be any quicker)…
Use the Range Tool, then select a Range between the Locators, then that same default Key Command Alt+Space.
Frustratingly enough, there is a Key Command for Cycle to follow Range Selection, but not the converse (Range Selection to follow Cycle) I can only guess that this is because a Range Selection has also a vertical domain in addition to the horizontal one.
Yes. Really. That usually just isn’t quick enough. I need an exactness that is time-consuming to achieve with the selection tool (I’d have to zoom in, and get the range in just the right location at both ends). So a play button to play between the L and R markers would be extremely convenient. After all, they are what determines the audio mixdown. (Sometimes I do have to use the selection tool for this, but it’s always a pain, and it would so much more convenient for my purposes to have a dedicated command.)
In the context I’m working in, there usually isn’t a Part that encompasses the desired range.