Why Not Include Month/Day?
By convention, when we cite an article, we almost always just use the year, since that’s a short enough chunk of time that most reference lists will not have duplicate short citations. When there are duplicates, we often use letters (e.g.) “Smith et al. (2018a)” and “Smith et al. (2018b)”. There usually isn’t a need to know the date more precisely.
What Is Included
Most (or all?) journals I’ve encountered do report this information: dates of initial submission, acceptance and publication. As others have said, preprint manuscripts are not the published version of the paper and therefore have no need to include this information.
There are cases, less frequent in the past century, wherein manuscripts have been published decades after they were written, or even posthumously. Often this was because the author thought their writings would be unpopular or even dangerous, so they waited for a time when society would be more receptive. In such cases, I would expect this information to be mentioned somewhere near the paper — perhaps in a preface section, an editor’s or publisher’s note, or — if nowhere else, on Wikipedia.
When In Doubt...
In modern academia, papers tend to be published soon after they are written — usually within a year or two of the paper being submitted. Previous versions of the paper (e.g. submissions to journals that did not accept that manuscript) don’t really count because they do not represent the manuscript’s final form. This is in part because science is a competitive venture, and scientists want their findings to be available to those who would use them as soon as possible.
It’s usually a safe bet to assume that a paper was written within a year or two of the publication date — unless you have reason to suspect otherwise, in which case you can usually do some additional digging to find out.