Hey Alan- you guys did manage to hit a serious stretch of poor weather, which can happen at any time on Denali.
I've started up Denali in every week of the season from May 1 - June 26th over the years and have watched weather trends on the mountain for the past 18 years in the hopes of determining when the "best" time to climb might be. I've not seen any benefits to any time of the conventional season that create a compelling reason to choose one time over another.
ExciteableBoy was on the right track, in my opinion, but I'd say there are three sub-seasons:
Early (before May 7):
-colder (in the last few years, but not from 1998 - 2003 -ish)
-less snow up high
-probably windpacked/blue ice above 15,000'
-not sure the weather is any more or less stable
-you'll need to build your camps, but this isn't necessarily a bad thing...
-lower glacier may or may not have any lower crevasse hazard (really depends on the year!)
Mid (peak) season (May 7 - June 10):
-generally warmer down low, getting HOT as you get into June
-snow will start falling up high and can be considerable by late May, even to the point of delaying your move to high camp due to avvy hazard
-the majority of climbers for the season will be on the mountain (historic peak week starts +/- May 21- with 275 climbers one year!)
-mixed bag of weather, like always up there
-you might move into camps, saving effort, but the camps might be gnarly...
-lower glacier is generally in good shape, but not always!
Late season (after mid-June):
-Can be very hot up to 14K and can rain on the lower glacier
-it can snow a lot up high, making for great travel, if you're not breaking trail, but also avvy hazard
-fewer climbers, but many more than in April
-generally lots of camps to choose from, so pick and choose
-lower glacier can be a minefield or can be in decent shape. You might need to wait a day or two for a freeze before traveling to KIA
The human variable is something to definitely consider. First of all, wrap your head around the fact that the West Buttress route is not a wilderness experience unless you head up in April. Embrace the social aspects of all the other climbers, as you are all in it together.
If 200+ climbers fly on in a given week and you have five days of bad weather, they'll get stacked up somewhere. We've seen teams turn back on summit day because they were too far back "in line" on the Autobahn, and it would have taken them 3-4 hours to climb to Denali Pass, a 1.5 - 2 hour jaunt.
Guide friends really like the early season time frame, but you need a really good day to summit, as it can be really cold up high. It's not ALWAYS cold up there, and we had a team summit from the north on May 1 a few years back, on a relatively warm day due to an inversion that settled over the mountain.
Later is better if you want to ski on the upper mountain. This only pertains to making turns, as capable skiers should consider traveling on skis at any point of the season.
When considering advice, I'd only caution you that anyone can have a relatively easy or hard time or two on Denali, and come away thinking that all the hype is just that (easy or hard). Folks who have seen the mountain in all its conditions might have different opinions on aspects of the experience.
To echo Jesus above- go do it! It is a beautiful, amazing place that anyone with a love of big, cold mountains needs to check out.
Ultimately- have fun up there!