This isn't about whether you or any other reconer is competent.
Simply put, a pair of JBL L166 speakers with excellent original woofers / original drivers is worth more than a pair of JBL L166 speakers with aftermarket drivers/diaphragms/cones. OTOH, if the speakers have rough cabinets, non-original tweeters, no original grilles it really doesn't much matter what you put in the woofer baskets for value.
The L166 is a very good speaker, one of my favorite classic JBLs, but its value to me and other collectors is as an original, not for its ultimate SQ which will be less accurate than for example a pair of new Studio 590s or many other lesser-known used speakers. To retain their value they need to have the original parts.
Maybe you're willing to buy speakers without original parts, but if so I suspect you're not the guy who was the backup bidder on my $3,350 L100As last year, you are more likely one of those who might pay $500 for a flip. The guy who bought them, or any other of my well-preserved collection of JBLs and others is a guy who wants only original.
This is not a concept unique to JBL or to audio, there's original and there's everything else in a collectibles market.
If the OP were interested in reconing the speakers I would suggest finding a pair of baskets that are already without cones or with damaged cones to have reconed, and set these woofers aside for the later option of reuniting them with the original boxes and the best value to a collector.