Using gBuffer from rgeos with correct projection

I want to show 15 mile radius circles around points in a map using gBuffer. As far as I can tell I have the points and the map in the same projection, but when I produce the circles on the map, they are too large. Here is my code. The tigerline files for the state and counties can be found at https://www.census.gov/cgi-bin/geo/shapefiles/index.php.

library(tidyverse)
library(rgdal)
library(rgeos)
library(ggplot2)

state <- readOGR('C:\\Users\\Mesonet\\Desktop\\map_folder\\tl_2020_us_state\\tl_2020_us_state.shp')
state <- state[which(state$STATEFP == '46'),]
state <- spTransform(state, CRS("+init=epsg:3857"))

counties <- readOGR('C:\\Users\\Mesonet\\Desktop\\map_folder\\tl_2020_us_county\\tl_2020_us_county.shp')
counties <- counties[which(counties$STATEFP == '46'),]
counties <- spTransform(counties, CRS("+init=epsg:3857"))

sites <- data.frame(Lon = c(-98.1096,-98.27935), Lat = c(43.9029, 43.717258))
coordinates(sites) <- ~Lon + Lat
proj4string(sites) <- CRS("+proj=longlat")
sites <- spTransform(sites, CRS = CRS("+init=epsg:3857"))

# Miles to meters conversion
mile2meter <- function(x){x * 1609.344}

# Buffer creation
site_buffer <- gBuffer(sites, width = mile2meter(15))

png('C:\\Users\\Mesonet\\Desktop\\map_folder\\new_test.png', height = 3000, width = 42*100, res = 100)
ggplot() + geom_path(counties, mapping = aes(x = long, y = lat, group = group), size = 1.75, 
  alpha = 0.45, col = 'darkgreen') + geom_path(state, mapping = aes(x = long, y = lat, group = 
  group), size = 0.8) + theme(axis.text = element_blank()) + geom_polygon(site_buffer, mapping
  = aes(x = long, y = lat, group = group), fill = '#0000FF', alpha = 1, size = 2)
dev.off()

These two locations are 15.35 miles apart, but the plot shows two circles that overlap each other by a couple miles. I can't figure out why, since from what I can see everything is in the same projection, but I might be wrong. Thank you.

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