Some of the advantages and problems associated with tail-mounted engines have been listed here above by some of you. Not everything is against the tail-mounted engines.
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There are advantages to have rear-mounted engines, such as reduced asymmetric thrust, and wing profiles favoring better aerodynamics, and noise levels being lower in the cabin... Despite wing-mounted engines, the 747 engines still require a tall ladder for oil servicing. So... maybe only the 737 and A-320 have convenient engine location for line maintenance.
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While tail-mounted engines benefit less damage on airports with foreign object ingestion, it could be the opposite in flight. When I flew the 727 (or the Learjets), we could not consider "deicing" the wing in flight after ice accumulation. It had to be "anti-icing"
prior reaching icing conditions, in fear of engines becoming ice blenders for your next "Margarita" or "Caipirinha" cocktails... Was so critical on the 727, that the VHF antenna (a "blade antenna") mounted on top of the fuselage was anti iced, in fear of ice ingestion into nš 2 engine.
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The other problem is bringing the fuel from wing tanks, to the engines. This requires long pipes (which may leak, therefore they are shrouded), and all this is extra weight. The T-tail associated with rear engines requires extra structural weight of the tail... And bad stall characteristics of T-tail airplanes required "stick pushers" for certification.
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You say that tail mounted engines make it easy to control an "engine-out" due to reduced control inputs...
not true... In the Lear 24B, VmcG is 108 KIAS and requires
full rudder input to continue the takeoff. The "multi-engine rating" obtained in a Learjet is not "center-line thrust only"... But the 727 is rather mild-mannered, and is "center-line-thrust"...
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Further, in the 727, I remember that on some landings, the use of reversers from nš 1 and 3 engines often blanked (or disturbed) the air over the rudder surface, making the rudder controls somewhat ineffective, for crosswind corrections. Finally, if an engine blows-up to pieces, I do not feel too good about the fact that it is very close to the tail surfaces...
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And to those of you telling us that the VC-10 is a nice looking airplane, we all have our preferences... So, my favorite airliner is the SE-210 Caravelle for the looks.
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Happy contrails