When I’m talking about “some corners of our country”, I’m talking about corners that I’ve personally seen myself. It’s not subtle when, e.g., the pool attendant lets me in the municipal pool without a second glance but stop the black mother behind me and pointedly demand she prove her residence.
Outright racial discrimination at a community pool is illegal. "Just itching to break out the noose" isn't really a thing.
What happens instead is something more like https://www.nbcnews.com/news/nbcblk/black-family-kicked-out-... where a white woman asked for the police to come because two black swimmers were talking to each other across her lane.
I don't know how to measure that prevalence, in any useful sense. There are also confounding factors as the history of segregation means black parents - who didn't learn to swim - don't think it's fun to take the kids to the pool, nor can teach the kids to swim.
It's certainly not low enough that it can be ignored. There's almost certainly research papers which address your question, if you go looking for them.
The third prevalence, which thebooktocome points out, is how public pools closed down, to prevent mandatory desegregation. Instead, whites started or joined private clubs, where racial discrimination is legal.
How many private pool are in your small rural part of the country? Even without explicit racial bias, are the membership fees high enough that the strong correlation between race and income results in an implicit racial bias?
For a deeper coverage, see "Still Drowning in Segregation: Limits of Law in Post-Civil Rights America", https://scholarship.law.umn.edu/cgi/viewcontent.cgi?article=... , which starts with this example:
] The absence of swimming pool access in many cities throughout the nation explains why in 2009 the Philadelphia-based Creative Steps Day Camp paid the Valley Swim Club, a private facility in Huntington Valley, Pennsylvania, a $1,950 membership fee for weekly access for their campers to the club swimming pool." According to newspaper reports, when some of the approximately sixty predominately black and Hispanic day campers "got in the pool[,] all of the Caucasian children immediately exited the pool."9 Some day campers reported overhearing "club members asking why so many black children were in the pool ... [and] using racial slurs while they were swimming."
] ... The swimming pool at issue in the Valley Swim Club incident, however, was private. After Brown and the closing of government-operated pools, many whites fled to privately-run pools.
That specific case was settled: https://www.justice.gov/opa/pr/justice-department-settles-ra... . The pool went bankrupt.
Still, that's one of the corners thebooktocome mentioned, and that paper should be a good starting point for you to find research on the prevalence.
Another might be "Navigating Rough Waters: Public Swimming Pools, Discrimination, and the Law" at https://scholarworks.bgsu.edu/cgi/viewcontent.cgi?article=14... .
Oh, right! and the 2007 book "Contested Waters: A Social History of Swimming Pools in America". I had forgotten about that one. Google Scholar has some 380 papers citing the book. The book is at https://archive.org/details/contestedwaterss0000wilt .