Braulio009
I am sorry to hear about the loss of your passport. Did you contact the Chilean embassy in Italy and inform them about the loss? Have you informed the Italian police, too?
Please kindly note, we are unable to give you a legally binding answer. Only Swiss immigration / border control will know. However, having worked at the airport, I know that Chilean citizens do not need a Schengen visa. Since you are already in Rome, you already entered the Schengen area, so I do not see any issue travelling to Switzerland with your ID only. However, to return to Chile, you need your passport or a supporting document / attestation by your embassy that you can return without a passport.
Please clarify with the embassy what measurements you have to take before leaving Italy.
Kind regards,
Roland