I am not sure if it gettting enough sun then. Sometimes indicrect light or dappled sun can help. About 5 hours of sun would be ok. The one that I am growing here HAS to get less than 5 hours because the leaves will fry otherwise during the hot summer months so I expect bloomage to be affected by that. Can you possibly raise the canopy of the other tree that is nearby?
Another concern is the moisture issues. Most lilacs develop flower buds on old wood (the exception being rebloomers). One possibility worth considering is moisture issues that cause the tree to abort the flower buds. This occurs when the tree can no longer support flower buds due to the lack of moisture. Since you do not water in Fall and Winter, this could be a problem that gets compounded if it has pests that stress the shrub. Have your three years in that house been dry in Fall and Winter? A minor solution to consider is to mulch the area around the tree well with 2-3" of mulch. But you ought to consider some source of moisture during the dry Fall/Winter. Have you considered drip irrigation or colleting water in rain barrels for the lilac?
Another possibility.... How is the soil in your area? Have you had a soil test done ever? Lilacs prefer alkaline soil so I was wondering if you had any mineral deficiencies. You can check for soil alkalinity (or acidity) using a soil pH Kit. Some of the kits can also alert you to a phosphorus deficiency, which could cause lack of blooms too.