I would certainly use periods, looks clearer and otherwise you could end up with eg on your face.

sorry.

To answer the original question - yes, it bothers me.

#caller - "Good morning, is this Jane Smith?" / "Can I please speak with Ms Smith?"

# Jane Smith - "Speaking."

Problem solved, thank you very much.
I can't believe this argument has gone on for so long.

Yes there's a difference between what people speak now and sounds acceptable, and what the rules say sounded acceptable once. One of the joys of English is that it is fluid and not so rigid and stuck behind grammar rules... see split infinitives and prepositions on the end of sentences, and various others (probably all with posts as long as this one).

Say what you want on the phone, either you'll sound normal, or pretentious, or dumb depending on what side of this argument the other person believes in... either way, it shouldn't cause a problem.