I've had it happen when I'm traveling, and my primary card suddenly stops working. I always carry backups for this reason, but it's still disruptive and potentially embarrassing.
Also if I have a card on file for recurring payments or repeat orders, I have to go find all of those places and update it every time the card gets replaced. When it happens once, it's not so bad. When it happens two or three times a year, it's a headache. I finally took to having a single card that is used for ONLY recurring payments, with the hopes that it wouldn't get compromised and when the others do I wouldn't have to go through this whole exercise again. So far so good.
The bottom line is yeah, it doesn't cost me money, but it does cost me stress and time. I have better things to do with my time and energy than cleaning up after yet another compromise that happened through no fault of my own.
This is a sound plan. Kudos!
For non-recurring payments, I've taken this a step further: I use a debit card for online purchases backed by an account with just a few dollars in it. When I wish to make a purchase, I move the funds for that purchase into that account.
I would rather have a fraudulent purchase be declined than to deal with the hassle -however small- of disputing a charge.
Also most recurring billers use account updater so if your bank supports it, the billers will get the new card #.
If the thief is slick he / she may make small charges that you wouldn't immediately notice or at all if you're sharing the card with someone else (e.g. a significant other). It's also a HUGE pain in the ass.