I had a similar experience w/ '97 E320. I noticed that there was no momentary fuel pump sound before the motor was cranked - and it cranked vigorously, no battery weakness. So I began to look why no fuel pump sound.
I soon found the cause. On the battery's + terminal is a Christmas tree of wires - the regular heavy cable and its clamp and attached to the clamping bolt were three additional red wires - one of which powers the fuel pump.
The battery was about 6 months old and must have leaked or been over filled because the + terminal was buried by corrosion - corrosion that literally "ate" the 6 mm bolt for the three additional wires. They fell off the battery when I touched them!
The repair was to remove the battery for replacement and clean and soak the affected clamps and wires in dilute baking soda and water solution several times to remove the acid. The three loose wires were added to the clamping bolt which I replaced with a 6 mm bolt (8 mm original) so the wires would fit on to it. Everything was greased and reassembled and works fine. Incidently the M-B replacement harness (to get the original Christmas tree arrangement costs $99 - you see why I redesigned the connection).
If your battery is OK I'd suspect the fuel pump or the immobilized if there is one, try another key.