You say you pushed it. You mean you literally pushed it by hand?
I am wondering if you have a loose heat shield somewhere. The 300E had a most annoying rattling sound, rather high pitched. It turned out to be a loose C clamp on the exhaust system, held in place by the bracket mounted on the transmission.
What was odd and made me think of your car is it wouldn't do it all this time.
'03 E320 Wagon-Sold
'95 E320 Wagon-Went to Ex
'93 190E 2.6-Wrecked
'91 300E-Went to Ex
'65 911 Coupe (#302580)