Yes, those get squished-stretched, if you are using anything but cube or quad, and the local axis orientation matter too. This would have been ok for the Wood textured planks or beams, not so much for other textures.. Local makes it almost like UV-based, deforms with scale. Besides, every object with that tex will have the same pattern.
For non-uniform scaling I'd still suggest using several versions of non-Local textures for various aspect ratios, which you can convert to 2D textures.
All said, not completely clear what you are after - in your image I see what happens, but I might only guess what it should look like. Still, I suspect it is impossible that easy, unless you are ready to write own shader. Might be tempting to apply one shader/texture to everything and expect to do all the dirty job. 😉 Here it sounds something to do with shading switches, yet there is no shading switch for the placement matrices.
I haven't used those, but you might be possibly after Texture References. Try, succeed.
Also consult Arnold's documentation, as it might have own tricks and limitations.
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Maya2019.1 @ Windows10 & GeForce GTX1080Ti
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