I've tried selecting an Armature in Pose Mode, switched to Object Mode, and then Weight Paint mode, but it still doesn't allow me to select bones, among a plethora of other solutions. I'm not sure why, but for some reason it just doesn't work for me. Can anyone help me figure out what I'm doing wrong?
You need to parent the mesh to the armature first to setup the weights, this is done with Lock Object Modes checked.
In Object Mode, select the mesh, Shift select armature, then Ctrl + P to parent and choose Armature Deform then select any sub-option, like With Automatic Weights.
You should now be able to Ctrl + left mouse on the bones in weight paint mode but only if you have selected the armature first, then Shift selected the mesh, and entered Weight Paint mode.
Select armature - go in pose mod - shift + select mesh - go in weight paint - select bone with ctrl + left click.
Edit: video with your file
One possible reason of this issue found in Edit > Preferences > Keymap. Go to Keymap and select Key-Binding mode of search. Then type "ctrl left mouse" in the search input field. Look for the result weight paint section, here are 3 values by default. Delete the Weight Paint - ctrl left mouse line with the little x button. Save preferences, and now Ctrl+LMB should work to bone select.