B767 CREW OVERSHOT THE TURN TO FINAL WHILE EXECUTING THE VOR-DME APCH TO RWY 22L AT JFK.

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NON PRECISION APCH AT JFK. WX AT JFK WAS 1800 FT OVCST. VOR/DME RWY 22L APCH IN USE. FO FLYING HAD MAP DISPLAY. CAPT'S DISPLAY WAS EXPANDED VOR. GIVEN HDG TO INTERCEPT FINAL APCH COURSE. LNAV CAPTURED BUT VOR SHOWED FULL SCALE DEFLECTION. WENT BACK TO HDG SELECT. VOR NEEDLE CAME ALIVE VERY QUICKLY. IT WAS LIKE THE NEEDLE MAY HAVE BEEN STUCK IN THE OLD DAYS. WE WENT THROUGH THE FINAL APCH COURSE SLIGHTLY. WE CAPTURED COURSE AND WENT BACK TO LNAV. TWR CTLR ASKED IF WE WERE DOING THE ILS OR VOR APCH AND WE SAID VOR. HE SAID WE WERE R OF COURSE AND WE AGREED WE WERE SLIGHTLY R AND CORRECTING. NORMAL LNDG ACCOMPLISHED. TWR SAID THEY HAD NO PROB WITH IT; BUT APCH CTL WANTED TO MAKE SURE WE WERE ON THE CORRECT APCH. WE FOLLOWED THE BOOK PROC CORRECTLY WITH BACKING UP WITH RAW DATA. THE VOR NEEDLE WAS JUST A LITTLE SLOW COMING IN. COMMENT: IN THIS DAY AND AGE; THERE IS NO REASON TO BE DOING A NON PRECISION APCH WHEN THERE IS AN ILS AVAILABLE. NOISE ABATEMENT SHOULD NOT TAKE PRECEDENT OVER SAFETY; ESPECIALLY IN A WDB ACFT.