I would imagine this is by design as, given you are using ‘add edge loop’ on exit from finishing the add, your newly added loops are selected, so edge mode would be the logical mode to be in.
However, if you leave the edges selected and switch to vertex select mode, the vertices of those edges would still be selected for you to manipulate in that mode. For face mode, it would depend on how many edge loops were added. To have the faces selected, you would first switch to vertex mode, then face mode. Just switching to face mode would not select any faces, nor would it if you only add one edge loop.
Hope this helps.