Resample raster

It is pretty standard practice to aggregate land cover maps into layers of %cover. I.e you should aim to produce 13 layers, each being something like %cover in that grid cell. Doing this allows you to reduce the resolution while retaining as much information as possible. N.B if you require a different summary statistic than %, should be easy to adapt the following method to whatever statistic you want, by changing the fun = function in aggregate.

The following method is pretty fast (it takes just a few seconds on my laptop to process raster with 100 million cells):

First, let's create some dummy rasters to use

Nhr <- 1e4 # resolution of high-res raster
Nlr <- 333 # resolution of low-res raster
r.hr <- raster(ncols=Nhr, nrows=Nhr)
r.lr <- raster(ncols=Nlr, nrows=Nlr)
r.hr[] <- sample(1:13, Nhr^2, replace=T)

Now, we begin by aggregating the high res raster to almost the same resolution as the low res one (to nearest integer number of cells). Each resulting layer contains the fraction of area within that cell in which value of original raster is N.

Nratio <- as.integer(Nhr/Nlr) # ratio of high to low resolutions, to nearest integer value for aggregation
layer1 <- aggregate(r.hr, Nratio, fun=function(x, na.rm=T) {mean(x==1, na.rm=na.rm)})
layer2 <- aggregate(r.hr, Nratio, fun=function(x, na.rm=T) {mean(x==2, na.rm=na.rm)})

And finally, resample low res raster to the desired resolution

layer1 <- resample(layer1, r.lr, method = "ngb") 
layer2 <- resample(layer2, r.lr, method = "ngb")

repeat for each layer, and build your layers into a stack or a multi-band raster

r_hr <- raster(nrow=70, ncol=70) #High resolution raster with categorical data
set.seed(0)
r_hr[] <- round(runif(1:ncell(r_hr), 1, 5))
r_lr <- raster(nrow=6, ncol=6)

r_hr
#class       : RasterLayer 
#dimensions  : 70, 70, 4900  (nrow, ncol, ncell)
#resolution  : 5.142857, 2.571429  (x, y)
#extent      : -180, 180, -90, 90  (xmin, xmax, ymin, ymax)
#coord. ref. : +proj=longlat +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0 
#data source : in memory
#names       : layer 
#values      : 1, 5  (min, max)

r_lr
#class       : RasterLayer 
#dimensions  : 6, 6, 36  (nrow, ncol, ncell)
#resolution  : 60, 30  (x, y)
#extent      : -180, 180, -90, 90  (xmin, xmax, ymin, ymax)
#coord. ref. : +proj=longlat +datum=WGS84 +ellps=WGS84 +towgs84=0,0,0 

Direct aggregate is not possible, because 70/6 is not an integer.

dim(r_hr)[1:2] / dim(r_lr)[1:2]
#[1] 11.66667 11.66667

Nearest neighbor resampling is not a good idea either as the results would be arbitrary.

Here is a by layer approach that you suggested and dww also showed already.

b <- layerize(r_hr)
fact <- round(dim(r_hr)[1:2] / dim(r_lr)[1:2])
a <- aggregate(b, fact)
x <- resample(a, r_lr)

Now you have proportions. If you want a single class you could do

y <- which.max(x)

In that case, another approach would be to aggregate the classes

ag <- aggregate(r_hr, fact, modal) 
agx <- resample(ag, r_lr, method='ngb')

Note that agx and y are the same. But they can both be problematic as you might have cells with 5 classes with each about 20%, making it rather unreasonable to pick one winner.