I find myself having to use the "Rest" checkbox quite often, so hovering over it, the tooltip reads that holding the tilde (`) key while clicking will insert a rest. This didn't work for me.
Anyone else having this issue?

Try Shift+tilde. I’m not at a PC to check, but there’s another key needed. Shift or maybe Alt or Ctrl?

Holding the tilde key down while left-clicking works, however, you need to have some actual notes on the stave also. BiaB notation system is a little unique. A longer note gets inserted, and its length is automatically shortened by a following note placement. Try placing two notes on a bar and then inserting a rest in between the two (tilde-click). It should work. (You need to be in Edit mode also!)

That’s a good point: you must be in the Editable Notation Mode. Also, could you tell us whether this is BIAB for Mac or Windows, and is it version 2018?

Yes, to affirm, you cannot simply place a rest on a bar unless there are accompanying notes - except if the bar has no notation you can place the rest on the very first beat of an empty bar, which will produce a semibreve rest (4 beats) for the entire bar - (presuming the time signature is 4/4).

As Matt pointed out, it's also helpful to advise exactly what version you are using.