I have a new 597. It has the newest Circle10 magazine. I stripped down and cleaned the rifle before its first. I did not take the magazine apart to do anything to it. I have had no extraction issues, but like so many Failure to Feed is another thing altogether. So far I have tried CCI and Winchester ammo, with the CCI being more consistent.
My question is that with the FTF, when you pull the trigger and you hear that "click", does that really wind up being the same as dry firing? And if so, how long can you really expect the firing pin to last?

No, not the same as dry fire...as the firing pin is still hitting the primer of the cartridge in the receiver.

My concern is that the cartridge is not being fed into the chamber, so no rim to strike.

In 2008 I took my son to a nearby Cabelas where they had a Remington 597 .22 WMR semi-auto with synthetic stock. He encouraged me to get it, so I purchased it right away he was the one that actually wanted it.
It was great, but did have probs with FTF and FTE.
I read that these causes were because of the magazines and that Remington had updated the original mags TWICE. I bought the latest version, but still was having probs up until last year when a friend on another board told me that I should get a Quartsen Extractor. I was game and ordered one then installed it. It worked GREAT!
The rifle has a 20" barrel. Manufactured in 2000.
Installed Bushnell Trophy Red/Green Dot Scope 1 x 28 on top of a picatinny rail.
Shoots great and is a lot of fun.
Excellent for starting all the young kids on, in the family.

UPDATE: January 14, 2021...Just before the Great Pandemic of 2020, I got some v.4? black plastic magazines for my Rem. 597M and they worked just fine. I think with the combo of the Quartsen Extractor and the newest mags, this rifle is just down right awesome. I am very lucky I ran across it at Cabela's Used Gun Library in Reno.
UPDATE: January 24, 2021...While surfin' the internet, I just found the black plastic magazines for the Rem. 597M were the OEM mags. Don't know why, but they switched to the metal ones which don't work very well.