Circular vs Linear Polarizer on m4/3 Camera

Here is what I've learned, what I bought, and why I bought it.

I bought a linear polarizer because it will work with my mirrorless, contrast detection M10ii and M5ii. Linear will work better than circular because linear polarizes more strongly. Normally linear costs way less than circular, but not in this case since I bought Heliopan linear which costs as much as a good Hoya circular.

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The only brands of linear polarizers I could find are made by Heliopan, Tiffen, and Hoya.

Hoya doesn't make size 37mm linear polarizer, or at least none available in that size.

Hoya and Tiffen linear polarizers are uncoated.

Heliopan was the only brand I could find for a 37mm multi-coated linear polarizer.

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I bought Heliopan linear (multi-coated) polarizer in size 37mm and 46mm for $69 each from BHPhoto.

Heliopan was the only brand of linear polarizer (that I could find) with anti-reflective lens coating on each side.

Heliopan linear polarizer is rated 5 stars by users at BHPhoto.

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The other good, high-end, linear polarizer I found was Hoya for $22 each. It does not have any lens coatings, but is rated 5 stars by users at BHPhoto. I've had excellent results from Hoya in the past and would have bought it, but it didn't have any lens coatings. I'm sure it's the best linear polarizer for the money, but it's not available in size 37mm, which is another reason I didn't buy Hoya.

The lack of lens coatings didn't matter much back in my film days when I got excellent results from Hoya and Tiffen uncoated linear polarizers. However, I'm not sure if that same applies to digital. Would the lack of lens coatings be a problem for digital? I don't know, which is why I went for the coated Heliopan at way more cost.

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Tiffen linear polarizer was available for $19, but has no lens coatings. It's rated 4 stars by BHPhoto customers. I didn't buy it because I think the Hoya and Heliopan are better. However, in my college days (using film) I used a Tiffen polarizer and it gave excellent results. It was great with film. It's probably good with digital too. But I can afford better now. Hoya is better for only $2 more. Heliopan is much better for $50 more.

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Though my question stands... Does lens coating (or lack of) on a linear polarizer matter more for digital than for film? Cause on film I was getting great results with cheap uncoated linear polarizers like Tiffen and Hoya.