This looks like a memory or storage issue. A 1 by 1 m raster is 25 times larger than a 5 by 5 m raster, so this is a huge change. You could have not enough RAM or the number that stores unique values is too small. Here are 2 workarounds (that need to be further developed).
1) use a subset
Create a regular vector grid of about 25 tiles
Make a 1 m buffer around each polygon
select the points by location inside each buffer
Create a vert to clip your raster
Get the count and mosaic the results (take the maximum in overlapping areas)
2) get the count without using the raster, then convert it to raster
If you have something like 1000 counts per pixel, the number of occupied cells should be "relatively" small. So you could focus on the points.
get a list of the coordinates of the points
convert the coordinates in meters and **round** your coordinates (all values that fall in the samme pixels will be equal, just check if your raster is shifted if you want to fit exactly within each pixel)
Loop on each pair of coordinate and increase their count by one (something like mydictionnary[(xcoordinate,ycoordinate)] += 1 , initialized at 1 if empty)
now you have your points with values of the counts, so you can continue to work with points or convert them into a 1 by 1 raster.