do moving coil cartridges have a finite life-span?

The person who told you that must have been in the business of selling cartridges. I've used MC and MM carts without suspension issues whose ages are measured in decades. However, the STYLUS does not have an infinite lifespan and a 4-year-old cart may well need retipping depending on how many hours of service it's had.

It is possible for the suspension to go bad, particularly from the rubber parts hardening, just sitting in the box. This is supposedly a bigger problem in urban areas with high pollution levels (ozone being the main culprit). But, most high quality cartridges don't exhibit such problems. I have heard some cartridges that are more than 20 years old that sound good.

As for stylus wear, a friend of mine has a Lyra cartridge that has well beyond 2,000 hours on it that still sounds very good. If the cartridge has a very high quality diamond and the records that it plays are kept clean, the stylus can last a very long time.

Many years ago, when I ownd Shure cartridges (Type IV, V), I noticed degradation with less than 400 hours of play;, while my MC cartridges have lasted many multiples of that amount of time. I have no idea as to whether this was because of a different grade of diamond used, whether microridge-type profiles are inherently longer lasting, or whether the suspension was more delicate with the Shures. Whatever the case, I am certainly pleased that the really expensive stuff seems to last a long time.

Eventually most rubber or polymers will harden, but i have a Van Den Hul Frog that is 12 yrs old and it still is very good with no apparent sag or issues.

I used to have Lyra Dorian. After about 6 years the cart was worn out and needed retipping. I traded it in for a Dynavector

It's my 12th year on Shelter 501(not MK2). It was already retipped twice as it's being used long hours daily.
Other than tip and suspension on the tip there's nothing else to die that quick.
I guess with proper service and retipping ontime it'll last another lifetime.