Will a 3x10 or 3x9 front derailleur work on a 6/7 speed chain?

Generally yes, the front mech can work acceptably with other sizes of chain.

Since you will have a thicker chain than expected, the front derailleur cage may need to be widened a little to reduce chain rub.

The actuation ratio is between your shifter and the chainrings, so a friction shifter will work perfectly, and an indexed shifter should be acceptable. If you're mixing shifter/chainrings then things could get odd but that's more likely above 10speed. Below 10 speed, the chainring spacing generally was the same.

Remember the front shifter is more of a "brute force" mechanism having to move the chain under tension, compared to the rear shifter whose chain is mostly slack. So the front shifter has to deal with higher pressures and tensions.

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I have an old MTB that was 15 speed, (3x5) which now has a 9 speed rear cassette/derailleur/chain. The front mech is still the original triple, on the original chainrings. All I had to do was close the cage a bit to improve shifting. I do use friction shifter on the left hand which makes trim very flexible.

Might I suggest looking at new-old-stock or used items from say the 1990s? Something like this (example only: https://www.ebay.com/p/1300427514 )