Issues resampling raster to the resolution of another raster

I am trying to take a population raster and resample+reproject it to match the shape and resolution of a precipitation raster.

Data Links:

Population Data: https://figshare.com/ndownloader/files/10257111

Precipitation Data: https://www.ncei.noaa.gov/data/nclimgrid-monthly/access/nclimgrid_prcp.nc

The Population Data is a series of rasters per decade of 5 different population models covering the continental US. If you simply select one of the rasters I can work out the rest (I have combined into a multiband raster anyways). For example if you use the pop_m4_2010 raster that would help. The resolution is 1x1km, and the projection is Albers Equal Area Conic NAD 83 ESRI:102003.

The Precipitation Data is a netcdf file covering monthly precipitation data for the continental US. The resolution is 5x5km and the projection is WGS84 EPSG:4326.

I converted the netcdf to tiff using the following code:

import xarray as xr 
import rioxarray as rio
prcp_file = xr.open_dataset('nclimgrid_prcp.nc')
prp = prcp_file['prcp']
prp = prp.rio.set_spatial_dims(x_dim='lon', y_dim='lat')
prp.rio.write_crs("epsg:4326", inplace=True)
prp.rio.to_raster('prp_raster.tiff')

I also used QGIS to open the population files (add raster layer, navigate into the downloaded folder for pop_m4_2010 and select the "w001001.adf" file). When I do this in a WGS84 project QGIS automatically appears to force reprojection but I am new to this so I am unsure if it is correct.

From this point I have tried several things to resample the population raster to match the 5×5 resolution of the precipitation raster.

1. In QGIS Processing Toolbox GRASS r.resample
2. In QGIS Processing Toolbox Raster Layer Zonal Statistics
3. In Python, honestly I have lost track of all of the different forum posts and tutorials I have followed on GDAL.Warp, Rasterio.Warp, affine transformations, rio.reproject_match, etc. Below are a few examples of the code attempts.

Many of these appear to work (particularly the rio.reproject_match seemed simple and effective). However, none of these appear to be working as intended. When I test the accuracy of the resulting population raster by passing zonal stats of a county vector shapefile the resulting sum of population in the area is either 0, or wildly inaccurate.

What am I doing wrong?

Reproject_Match:

import rioxarray # for the extension to load
import xarray

import matplotlib.pyplot as plt

%matplotlib inline

def print_raster(raster):
    print(
        f"shape: {raster.rio.shape}n"
        f"resolution: {raster.rio.resolution()}n"
        f"bounds: {raster.rio.bounds()}n"
        f"sum: {raster.sum().item()}n"
        f"CRS: {raster.rio.crs}n"
    )

xds = rioxarray.open_rasterio('pop_m4_2010.tif')
xds_match = rioxarray.open_rasterio('prp_raster.tiff')

fig, axes = plt.subplots(ncols=2, figsize=(12,4))
xds.plot(ax=axes[0])
xds_match.plot(ax=axes[1])
plt.draw()

print("Original Raster:n----------------n")
print_raster(xds)
print("Raster to Match:n----------------n")
print_raster(xds_match)

xds_repr_match = xds.rio.reproject_match(xds_match)

print("Reprojected Raster:n-------------------n")
print_raster(xds_repr_match)
print("Raster to Match:n----------------n")
print_raster(xds_match)

xds_repr_match.rio.to_raster("reproj_pop.tif")

Another way with Rasterio.Warp:

import rasterio
from rasterio.warp import calculate_default_transform, reproject, Resampling

#open source raster
srcRst =rasterio.open('pop_m4_2010.tif') 
print("source raster crs:")
print(srcRst.crs)

dstCrs = {'init': 'EPSG:4326'}
print("destination raster crs:")
print(dstCrs)

#calculate transform array and shape of reprojected raster
transform, width, height = calculate_default_transform(
        srcRst.crs, dstCrs, srcRst.width, srcRst.height, *srcRst.bounds)

print("transform array of source raster")
print(srcRst.transform)

print("transform array of destination raster")
print(transform)

#working of the meta for the destination raster
kwargs = srcRst.meta.copy()
kwargs.update({
        'crs': dstCrs,
        'transform': transform,
        'width': width,
        'height': height
    })

#open destination raster
dstRst = rasterio.open('pop_m4_2010_reproj4326.tif', 'w', **kwargs)

#reproject and save raster band data
for i in range(1, srcRst.count + 1):
    reproject(
        source=rasterio.band(srcRst, i),
        destination=rasterio.band(dstRst, i),
        #src_transform=srcRst.transform,
        src_crs=srcRst.crs,
        #dst_transform=transform,
        dst_crs=dstCrs,
        resampling=Resampling.bilinear)
    print(i)
    
#close destination raster
dstRst.close()

And here is a second attempt with Rasterio.Warp:

import rasterio
from rasterio.warp import calculate_default_transform, reproject, Resampling

prcp = rasterio.open('prp_raster.tiff', mode = 'r')

with rasterio.open('pop_m4_2010.tif') as dataset:
    # resample data to target shape
    data = dataset.read(out_shape=(dataset.count,prcp.height,prcp.width), resampling=Resampling.bilinear)

    # scale image transform
    transform = dataset.transform * dataset.transform.scale((dataset.width / data.shape[-1]),
                                                            (dataset.height / data.shape[-2]))
    
    # Register GDAL format drivers and configuration options with a
    # context manager.
    with rasterio.Env():

        profile = src.profile

        profile.update(
            dtype=rasterio.float32,
            count=1,
            compress='lzw')

        with rasterio.open('pop_m4_2010_resampledtoprcp.tif', 'w', **profile) as dst:
            dst.write(data.astype(rasterio.float32))