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.