Relays are magic - like electricity
I have learned from the many postings on this board that fuel pump relays are not always responsive to logic.
I had similar problems to yours on my 1990 560SEL. I took the cover off the relay and searched hard with a magnifying glass for cracks and could find none. After the problem resurfaced I sprung for a new relay and it's been fine since. About $110.
As a back-up, I carry the old relay in the trunk and also a 4 inch piece of stout insulated wire. In a pinch, you remove the relay and insert the wire in terminals 7 and 8 on the relay socket. (The #s are on the socket, not on the relay). The pump will run if it is OK and the engine will start, though you will lose the rev limiter which, I understand, is part of the relay, so go easy on the gas. The pump will continue to run as long as your jumper wire is in place, so remove it once you have limped home.
If you place your ammeter between terminals 7 & 8 you will not only hear the pump(s) run, but will also see the current draw. Mine draws 6.8 amps, which appears within spec per Steve's email above.
BTW, you will see two relays at the firewall on the driver's side - the fuel pump relay is the one that is NOT marked 'Klima' - that one is for the climate control. Also, the positions of the two relays were switched (in 1988 or so - can't swear to the date, it's in the Electrical Troubleshooting Manual) so it's not enough to say 'it's the one on the left').