do moving coil cartridges have a finite life-span?

I was told about 4 years before the suspensions go bad.
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.
A 4 year rule is in general wrong. The majority of MC can easily do MUCH more years. Before I got the cartridge bug I used a Takeda Miyabi for 10 years more or less daily and it was still ok when it found a new owner. But it was running with a correct designed Arm and the right AS, Azimuth and VTF.
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.
I reinstalled a JMAS MIT-1 which is ~30 years old and it sounds great. I had sent it last year to Andy Chong,who inspected it, cleaned the stylus (vdH) and reconditioned the suspension.