Fenders help in keeping the sand, salt, and grit from getting kicked up onto the chain.

But honestly short of replacing the whole bike the best solution is usually to go with a reliable cheaper chain then replace it every year.

I've experienced the exact opposite. The rear fender keeps the frame and your back clean but drop everything that would go elsewhere now drops to the drive train and everything below.

I thought that for a while, until I got caught in the rain on my "dry bike." Short shower and perhaps 5-10 miles on wet roads, and the chain looked worse than the rain bike's chain after a solid week of commuting (80-100 miles). I was convinced after that: fenders are the way to go.

It may all be relative, my bike usually comes home cleaner if it is actively raining. I sometimes use a mud guard on the lower frame tube and an "ass saver" shoved in the seat back but not all the time.

WRT chain, a good front fender - and a good mud flap - is more important than a rear. It should come down close to the ground. That keeps the rooster tail of road grime drawn up from the front wheel from hitting your chain and chainring. This is the one on my winter bike:

(Yes, it is a piece of an old boot. Good rubber, long lasting)
I use hot wax as a lube on this one, get about 500 km between applications. I’m alternating between two chains and will hit 5000 km at the next change. No obvious drivetrain performance yet. This is in Sweden, so there’s plenty of sanded/salted winter riding.
For various reasons, it’s the only fender-equipped bike I have that’s in order at the moment, so it has the summer wheels on in the pic.