I do not understand how you can get your results. I did nothing differently than you did, except added the -s_srsparameter. If that is not used your command is interpreted so that EPSG:3857 is the source image and that leads to error so I do not believe that you have really used your command literally like you wrote it into the question.
ERROR 4: EPSG:3857: No such file or directory
I used the images from your gdal.zip package.
Gdalinfo about the source images:
Driver: GTiff/GeoTIFF
Files: KEWX_L2_REF.tif
Size is 4000, 3500
Coordinate System is:
GEOGCS["WGS 84",
DATUM["WGS_1984",
SPHEROID["WGS 84",6378137,298.257223563,
AUTHORITY["EPSG","7030"]],
AUTHORITY["EPSG","6326"]],
PRIMEM["Greenwich",0],
UNIT["degree",0.0174532925199433],
AUTHORITY["EPSG","4326"]]
Origin = (-102.000000000000000,33.000000000000000)
Pixel Size = (0.002000000000000,-0.002000000000000)
Corner Coordinates:
Upper Left (-102.0000000, 33.0000000) (102d 0' 0.00"W, 33d 0' 0.00"N)
Lower Left (-102.0000000, 26.0000000) (102d 0' 0.00"W, 26d 0' 0.00"N)
Upper Right ( -94.0000000, 33.0000000) ( 94d 0' 0.00"W, 33d 0' 0.00"N)
Lower Right ( -94.0000000, 26.0000000) ( 94d 0' 0.00"W, 26d 0' 0.00"N)
Center ( -98.0000000, 29.5000000) ( 98d 0' 0.00"W, 29d30' 0.00"N)
Driver: GTiff/GeoTIFF
Files: KEWX_L2_VEL.tif
Size is 4000, 3500
Coordinate System is:
GEOGCS["WGS 84",
DATUM["WGS_1984",
SPHEROID["WGS 84",6378137,298.257223563,
AUTHORITY["EPSG","7030"]],
AUTHORITY["EPSG","6326"]],
PRIMEM["Greenwich",0],
UNIT["degree",0.0174532925199433],
AUTHORITY["EPSG","4326"]]
Origin = (-102.000000000000000,33.000000000000000)
Pixel Size = (0.002000000000000,-0.002000000000000)
Corner Coordinates:
Upper Left (-102.0000000, 33.0000000) (102d 0' 0.00"W, 33d 0' 0.00"N)
Lower Left (-102.0000000, 26.0000000) (102d 0' 0.00"W, 26d 0' 0.00"N)
Upper Right ( -94.0000000, 33.0000000) ( 94d 0' 0.00"W, 33d 0' 0.00"N)
Lower Right ( -94.0000000, 26.0000000) ( 94d 0' 0.00"W, 26d 0' 0.00"N)
Center ( -98.0000000, 29.5000000) ( 98d 0' 0.00"W, 29d30' 0.00"N)
Your commands which I re-used
gdalwarp -t_srs EPSG:3857 KEWX_L2_REF.tif KEWX_L2_REF-projected.tif
gdalwarp -t_srs EPSG:3857 KEWX_L2_REF.tif KEWX_L2_REF-projected.tif
Shortened gdalinfo about the resulting GeoTIFFs
Driver: GTiff/GeoTIFF
Files: KEWX_L2_REF-projected.tif
Size is 3746, 3770
Coordinate System is:
PROJCS["WGS 84 / Pseudo-Mercator",
....
AUTHORITY["EPSG","3857"]]
Origin = (-11354588.060913905501366,3895303.963393894489855)
Pixel Size = (237.710730685124162,-237.710730685124162)
Corner Coordinates:
Upper Left (-11354588.061, 3895303.963) (102d 0' 0.00"W, 33d 0' 0.00"N)
Lower Left (-11354588.061, 2999134.509) (102d 0' 0.00"W, 26d 0' 1.56"N)
Upper Right (-10464123.664, 3895303.963) ( 94d 0' 2.96"W, 33d 0' 0.00"N)
Lower Right (-10464123.664, 2999134.509) ( 94d 0' 2.96"W, 26d 0' 1.56"N)
Center (-10909355.862, 3447219.236) ( 98d 0' 1.48"W, 29d33'38.89"N)
Driver: GTiff/GeoTIFF
Files: KEWX_L2_VEL-projected.tif
Size is 3746, 3770
Coordinate System is:
PROJCS["WGS 84 / Pseudo-Mercator",
....
AUTHORITY["EPSG","3857"]]
Origin = (-11354588.060913905501366,3895303.963393894489855)
Pixel Size = (237.710730685124162,-237.710730685124162)
Corner Coordinates:
Upper Left (-11354588.061, 3895303.963) (102d 0' 0.00"W, 33d 0' 0.00"N)
Lower Left (-11354588.061, 2999134.509) (102d 0' 0.00"W, 26d 0' 1.56"N)
Upper Right (-10464123.664, 3895303.963) ( 94d 0' 2.96"W, 33d 0' 0.00"N)
Lower Right (-10464123.664, 2999134.509) ( 94d 0' 2.96"W, 26d 0' 1.56"N)
Center (-10909355.862, 3447219.236) ( 98d 0' 1.48"W, 29d33'38.89"N)
-t_srs? Corner coordinates of the warped and rotated images are not very useful for judging if the georeferencing if right or wrong because they are the corners of the final image which contain now also nodata areas. See for example https://gis.stackexchange.com/questions/131465/gdal-rotate-dem/131502#131502. Have you opened both warped images together with for example QGIS? Do they match? – user30184 Sep 13 '18 at 09:26