Does GRE play as a cut-off point in one's Economics Phd application even though s/he has a strong profile including 4.0 GPA, strong recommendation letters, good SOP, and relevant research and teaching experience?
Yes, it does, at a large number of, although certainly not all, top economics departments in the United States. Indeed, they have a lot of applications to read from perfect applicants (at least on paper). Why should they give any exceptions?
But they do give exceptions. I don't know entirely your circumstances, and certainly mathematics isn't economics. But from my own experiences, having 1. written letters for people with bad mathematics subject test GRE scores and 2. being swayed when I've been on committee to admit people with bad mathematics subject test GRE scores, I would encourage at least one of your letter writers to contact someone at each of the schools you applied to to keep a lookout for your application/not throw it out immediately presuming a GRE cutoff, especially if other parts of your application are quite competitive for the schools you are applying/have applied to.
You can contact the admissions department at the schools you are interested in if their website does not explicitly indicate a cut-off. A strict cut-off is rare, but does exist. If you are not getting a straight answer from them, your only option is to just go for it. There is a lot of unpredictability in PhD admittance decisions.
In my case (social sciences PhD), I had a GRE score that was less strong compared to my GPA, and it was mostly a factor in whether or not I would be offered a scholarship. At the end of the day, my GPA and recommendation letters were strong enough and I was offered the funding.
In my experience with economics, it's all over the place. Some departments do seem to use it as a cutoff, but increasingly more seem to be coming around to using them only at the margins.
This especially holds true if you have good recommendation letters and research and teaching experience; it's difficult to get those things before you start your PhD. They likely make you stand out well above the crowd. Heck, it's not unusual for full PhD graduates on the job market to lack teaching experience.
If a GRE quant score is going to be a problem, I anticipate it would matter more at the most highly selective institutions. Places that only accept a tiny fraction of near-perfect students have few reasons to compromise on any attribute, and GRE scores are an incredibly convenient standardized way to compare everyone.
I've spoken with faculty on admissions committees at both large public research universities and small, selective private universities, and while you can't use my anecdotes as the whole story, the responses I get are invariably that they rely as little as possible on the GRE.