I do think that unless it's a VERY popular potential area of doctoral study with loads of superstar advisors, the applicant needs to refine the topic or other aspects of study a bit more if there are 13 doctoral programmes that he or she would apply to (not could apply to). I'm not all that impressed when I get an email with the equivalent of "I'd like to study quantitative economics at PhD level -- will you supervise me?"
On the other hand, I can think of twenty people at good programs who could supervise someone whose interests are molecular dynamics of the physical properties of alkanes. Give me an hour and I could probably come up with fifty names of people at programs worth attending.
This might be one of those examples where there are truths on one side of the arts and sciences divide that are falsehoods on the other side.
Fit tends to be very specific in my discipline, especially because there are so many potential choices of methodology and method, which don't suit everyone equally.
I would still think that a wise applicant would consider whether all thirteen potential programmes would be "a good fit beyond the general area that [OP] is interested in" -- questions such as what's the required RTA load, do I love/hate small towns, do I love/loathe hot/cold weather, what are the course requirements, how fast do people complete, what is the record on placements, is this school above/below my reach, etc. If 50 schools meet those criteria and the applicant is indiff erent between them, then 50 applications might be appropriate.
On the other hand, when I was running our PhD programme (ranked in the top 15 in the world, >100 enrolled students) and reading 300-400 applications per year, I did want to see more than a generic, mail-merged* application that could apply to half the supervisors across several subdisciplines, with no compelling reason for the student to be studying with us.
*And I do mean mail-merged literally -- some even along the lines of "I have always wanted to study for a doctorate at [insert name here, but give the name of main rival university]" or "My dream is to study [topic not supervisable at this university] with [person gone so long I had never heard of them]".