[...]you're better off using a decent resampling algorithm that doesn't require such stupidity in the first place.
I fully agree. It's amazing that after all these years there is still no option for a slower but much better behaved downsampling algorithm in Photoshop.
Bicubic Sharper is one of the best available options in Photoshop, especially if you don't do any sharpening after resampling.
I disagree on the 'one of the best' part. With regards to aliasing it is worse than regular bicubic (as implemented in Photoshop).
I did a number of comparisons several years ago that show the potential differences/risks:http://www.xs4all.nl/~bvdwolf/main/foto/down_sample/down_sample.htm
and for some less theoretical examples on a real image:http://www.xs4all.nl/~bvdwolf/main/foto/down_sample/example1.htm
Especially the aliasing of the sunscreens will be what probably can be seen in the OP's 'screen door' effect.