I would definately recommend keeping it and getting someone decent to prune it, ideally a tree surgeon. We had two in our last garden and they were very old but looked much better (the following year that is) after a good prune. I think lilacs are far too lovely to get rid of though I know what you mean about the base of the tree and suckers. But the suckers are easy enough to trim away, and you could grow a lovely climber up the tree to cover the straggly parts, ideally one which flowers at another time from the lilac, to give you flowers in succession, such as a scented rose (Iceberg flowers all summer) or a summer flowering clematis, or both even.
Oh dont listen to me. I have no real clue about LILACS, but seem to remember my sister saying this, and she is the editor of a gardening magazine, so is quite knowledgable. But, I could have misunderstood her.
Am not sure about the suckers - simply thought lilacs were prone to suckering as many trees are. You could check the BBC/RHS sites for more info. A good tree surgeon will quote you for pruning but will also offer advice about this.
I know what you mean about the tree, lilacs not the most beautiful when not in flower (though I think they look fine), hence my suggestion to fill it with repeat flowering climbers. I recently saw a photo of a sycamore (quite an ugly tree) with a lovely white climbing rose scrambling through it and it looked gorgeous. I had an evergreen clematis armandii growing through one of our lilacs in the previous garden which also looked great (scented flowers too).
IMO the flowers and their scent are worth the negatives.