You could use the exclude parameter of distance if you use a single raster. To do that, set the cells that overlap the target points to a value different from the rest of the raster:

library(terra)
# create dataset
r <- rast(ncols=36, nrows=18, crs="+proj=longlat +datum=WGS84")
r[1:200] <- 1
p2 <- vect(rbind(c(30,-30), c(25,40), c(-9,-3)), crs="+proj=longlat +datum=WGS84")

# set values of cells overlapping points to -1
r[cells(r,p2)[,2]] <- -1
dist <- distance(x = r, target = 1, exclude= NA)
plot(dist)