A B737 CREW FLEW AN UNSTABILIZED APPROACH TO A LANDING AFTER ATC KEPT THEM HIGH PRIOR TO THE ARRIVAL.

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LATE DSCNT FROM ATC INTO ZZZ (WHICH WE RECOGNIZED). THREW OUT THE GEAR AND SPDBRAKE VERY EARLY (AROUND 13000 FT) TO TRY AND GET BACK ON PROFILE TO RWY XX. ASKED FOR MANEUVERING ROOM TO E OF FINAL. TWR APPROVED; BUT ALSO OFFERED R DOWNWIND TO RWY YYL WHICH WE ACCEPTED. WE WENT FROM HIGH TO ON PROFILE. WE HAD SELECTED FLAPS 40 DEGS FOR THE RWY XX APCH; BUT RESELECTED 30 DEGS FOR THE APCH TO RWY YYL. THE FO (PF) EVEN NOTED THAT NOW HE WOULD HAVE TO ADD PWR. I QUICKLY THREW IN RWY YYL INTO THE FMC FOR THE VNAV GUIDANCE AND WE CONTINUED. TRANSITIONING FROM A LONG BASE TO FINAL WE BECAME HIGH BUT CONTINUED. I TOLD THE FO HE WAS HIGH AND CALLED SINK RATE AT 1000 FT (AIRSPD WAS FINE; PWR IN; BUT ABOVE PAPI). APCH WAS CONTINUED AND WE LANDED WELL BEYOND THE NORMAL TOUCHDOWN POINT. WE SLOWED AND EXITED THE RWY NORMALLY. I FELL INTO A HUMAN FACTORS TRAP. WHEN THE HIGH PROFILE TO RWY XX WAS RESOLVED; I LET MY GUARD DOWN. THINGS (MOMENTARILY) BECAME NORMAL AGAIN. I FAILED TO DIRECT A GAR WHEN WE REACHED 1000 FT AND WE WERE STILL WELL ABOVE GS. ADDITIONALLY; AFTER BRIEFING AND LOOKING AT 12000 FT OF RWY XX; AND THEN SEEING RWY YYL; WHICH LOOKS 12000 FT WITH THE OVERRUN; BUT IS ONLY 9000 FT WITHOUT THE OVERRUN; I ACCEPTED A POSSIBLE LONG LNDG. THIS WOULD HAVE BEEN WRONG EVEN IF THE RWY WAS 12000 FT. AS IT WAS; WE DID LAND LONG. BOTTOM-LINE; DIRECT THE GAR.