You went via FCO!? Well, therein lies the problem. Why not fly KL via AMS instead?

The agent in OTP lied to you. You should have insisted and/or asked to speak to someone more informed, such as a supervisor. You can too interline bags on SkyTeam airlines.

If a long flight such as FCO-DTW is delayed, the GAs in FCO can not help you. They're only responsible for that flight alone. A transfer desk would normally help you, but for a longhaul flight I would wait until I actually get there. There can be further delays (bad winds, storm on the way, immigration delays, etc) and you're better off waiting until you're in DTW to do this.