Since it is continuous data you should either use bilinear interpolation or cubic convolution. The advantages and disadvantages of each can be found here. https://desktop.arcgis.com/en/arcmap/10.3/tools/environments/resampling-method.htm
Normally when rendering an image, EE works at different scales for different zoom levels. The docs explain how this works. When you reproject, you force EE to work at the scale you specified, and depending on your zoom level, each map tile will contain more or less pixels. When zoomed out too far, each tile simply contain too many pixels for EE to handle . ...
You are probably passing other resampling algorithm than max when using the solution you linked. The code should look like:
from osgeo import gdal
gdal.Warp('resampled_image.tif', 'input_image.tif', xRes=32 yRes=32, resampleAlg=gdal.GRA_Max)
Another alternative is calling the command from Python using the subprocess module: