I picked my dad up from dropping a car off from detail half an hour ago and he bumped into an old friend with a Kia that had the EXACT same issue as the 300E.

He said the reason was because the fuel pump in the gas tank had a filter with little pellets to cleanse the fuel, but that filter broke and the pellets got sucked into the fuel pump and would clog it up. When he let the car sit it would start right back up because the pellets had backed off. What makes this even more plausible despite the fact he had 100% the exact same symptoms as I is that the gas tank in the 300E had been messed with by the PO and we had to realign it and move it around. I don't know if Mercedes used this "pellet" set up (probably not considering it was Kia lol) but it seems like that is it!

Where is the fuel pump located on the 300E? The fuel tank?

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76 MG Midget
78 Datsun 280Z
86 Peugeot 505
01 Subaru Outback

“Owning a car is like owning a dog. Every now and then, it craps on the carpet — but it’s still your dog.” - Jeremy Clarkson