How I'd put it: They are useful because the integral of the likelihoods is not infinitesimal. The probability that your yard stick measures 1.0000000000000 ya

How I'd put it: They are useful because the integral of the likelihoods is not infinitesimal. The probability that your yard stick measures 1.0000000000000 yards is basically zero, but the probability that it's within one inch of that is close to one. We generally prefer to use probability density functions with the property that most of the probability density is close to the maximum likelihood. So, in the yard stick example, the yard stick lengths are probably gaussian, so if check enough lengths, you'll get a mean (== length with the maximum likelihood) that approaches 1.00000000000 yards (you'd hope) with some small standard deviation (probably less than an inch).