Lilacs adore chilly winters. Winters that get cold and stay cold. Places like the coastal NW they grow and bloom fine for the most part, but I grew up in Spokane, WA and they were in an entirely different class there. Lilacs went berserk.
Still, to not get much bloom at all.....
Is it budding, just not blooming? What kind of fertilizer to you put around the garden? Think not just of what you are doing for the lilac, but for the garden surrounding it. 7-10 years is not too old for lilac branches. A shrub will do well to occasionally have its old branches replaced, but 7-10 years is in the prime of life for a lilac branch. Pruning will not necessarily encourage more blooms in lilac.
Lilacs do fine in acidic soil, but prefer it towards neutral or the alkaline side. But we have acidic soil where I live and they do all right (not to the standards of my childhood, though.)
If it looks like they *bud* nicely but don't *bloom* I would suspect your high desert springs. Sweden and Spokane might get cold winters and springs, but I imagine in the desert you will be seeing quite warm spring days with frigid cold nights. I think that would be confusing for lilacs, as it would with fruit trees and other shrubs.
But are you getting buds at all? If not many, I would first consider the cultural practices of the rest of the garden. Overwatering, overfertilizing. I seriously doubt that desert soils are going to be acidic. Too alkaline perhaps? (Too saline, perhaps, especially if you irrigate. But I think that would affect the plant as a whole.) All possibilities. My first guess in this line of thought would be too much nitrogen if the plant looks healthy ("overgrown") but isn't blooming. Sometimes young shrubs are slow to get going, but I haven't seen lilacs do this, but then I haven't seen lilacs in desert areas outside Eastern Washington.
For me, the answer lies in whether or not the buds it makes are able to blossom, or if the buds just aren't there at all.
Zones are funny. We live in the same USDA zone as you do, but I'm sure our springs and winters don't look much like yours at all! A better guide, I think, is the Sunset Western Garden Guide, which has 24 zones. I think I am in their zone 4.