Wrong, if your lawn is that bad, new turf is the only answer. You (like myself) have yourself to blame as most weeds are airborn - like the little spores we all call fairies!!! lawns need to be mowed twice per week to stop the fairies and other spores bedding into the grass. Believe me I know what I am talking about, just had new lawns laid front and back for the same bloody problem.
One the cat front, yes again, keep the lawn short.