Career achievements, Murray is obviously greater. That being said, Courier was the dominant player for a time, against great competition.
Also I believe that Courier's style of play was unsuited for the 90's playing conditions. Grass and a lot of hard courts were rewarding more aggressive players, clay was home of tiredless defenders. Aggressive baseliners had the advantage on slower hard courts only. Still he didn't only met success at the AO and also won RG, made finals at the USO, Wimbledon and the Master Cup. A player like him would be much more comfortable today (altough wouldn't necessarily have more success due to top-heavy competition).
Jim's peak was far greater. At one point in the early 90s he was a dominant and deserving number 1. Andy has never had that level of dominance on tour. Longevity and consistency obviously goes to Andy but it's far easier to have consistent results in today's game than it was in the 90s.
It's easier today to have consistent results across surfaces but it isn't to have consistent results on specific surfaces. Also longevity is something else, I don't see how it was harder to have longevity in the 90's in comparison with before or later. Courier had a very short peak, then quickly vanished. He is responsible for it and could have added to his tally of titles (only 23) in the second half of the 90's which was pretty open.
it's so overstated that you had top ranked clay court players skipping Wimbledon and fast court players skipping the French Open and half of the field skipping the Australian because it was just too far and didn't offer enough prize money.

Imagine that happening today.
But while it made it harder to have success across surfaces, didn't it make it much easier to have success on a specific surface? I mean, if half of the pros are clumsy on your favorite surface?
Imagine a player who is extremely proficient on fast courts. For such a player, winning Roland-Garros against guys like Bruguera or Muster will be very though, as he would need to tweak his game too much in order to be competitive. But on his surfaces, it's the contrary. Defenders have no hopes. Aggressive baseliners (Agassi, Courier, Kafelnikov) can make it but must overcome disadvantages. This guy need only to be better than the other good fast-courters to utterly dominate the fast surfaces. Such a guy could probably win a few half-dozen of Wimbledon, USO and Master Cups, and it would be easier for him to do it than for the greats who preceded him or succeeded him because, while they didn't need to tweak there game as much to compete on all surfaces, they had to actually face the best players in the world.
Murray is one of the most fortunate #1s we've had on the tour - half the reason he's there is because 2 of the big 4 were injured for half a season, if not more.
I kind of agree but Murray has been waiting behind these guys for nine years before they faded, and then he still had to win a slew of tournaments back-to-back to make it. Okay, he didn't face the best competition in those, but going on a tear is never easy. And still being able to go on a tear after nine years of top level tennis is even harder. He deserve respect for it.
Look, how comes Courier didn't use HIS good fortune to go on a rampage in the late-90's? There was plenty of space to be number 1 again, and some lesser players took advantage of it (Rios anyone?).