Right - I think this is what's at the heart of the original question.
I know they asked with a continuous example, but I don't interpret their question as limited to continuous cases, and I think it's easier to address using a discrete example, as we avoid the issue of each exact parameter having infinitesimal mass which occurs in a continuous setting.
Let's imagine the parameter we're trying to estimate is discrete and has, say, 500 different possible values.
Let's say the parameter can have the value of the integers between 1 and 500 and most of the mass is clustered in the middle between 230 and 270.
Given some data, it would actually be possible that MLE would come up with the exact value, say 250.
But maybe given the data, a range of values between 240 and 260 are also very plausible, so the likelihood of exactly 250 has a fairly low probability.
The original poster is confused, because they are basically saying, well, if the actual probability is so low, why is this MLE stuff useful?
You are pointing out they should really frame things in terms of a range and not a point estimate. You are right; but I think their question is still legitimate, because often in practice we do not give a range, and just give the maximum likelihood estimate of the parameter. (And also, separately, in a discrete parameter setting, specific parameter value could have substantial mass.)
So why is the MLE useful?
My answer would be, well, that's because for many posterior distributions, a lot of the probability mass will be near the MLE, if not exactly at it - so knowing the MLE is often useful, even if the probability of that exact value of the parameter is low.
I know they asked with a continuous example, but I don't interpret their question as limited to continuous cases, and I think it's easier to address using a discrete example, as we avoid the issue of each exact parameter having infinitesimal mass which occurs in a continuous setting.
Let's imagine the parameter we're trying to estimate is discrete and has, say, 500 different possible values.
Let's say the parameter can have the value of the integers between 1 and 500 and most of the mass is clustered in the middle between 230 and 270.
Given some data, it would actually be possible that MLE would come up with the exact value, say 250.
But maybe given the data, a range of values between 240 and 260 are also very plausible, so the likelihood of exactly 250 has a fairly low probability.
The original poster is confused, because they are basically saying, well, if the actual probability is so low, why is this MLE stuff useful?
You are pointing out they should really frame things in terms of a range and not a point estimate. You are right; but I think their question is still legitimate, because often in practice we do not give a range, and just give the maximum likelihood estimate of the parameter. (And also, separately, in a discrete parameter setting, specific parameter value could have substantial mass.)
So why is the MLE useful?
My answer would be, well, that's because for many posterior distributions, a lot of the probability mass will be near the MLE, if not exactly at it - so knowing the MLE is often useful, even if the probability of that exact value of the parameter is low.