If a picture is worth a thousand words, here's a few thousand words on why to have a red bridge and how traditional they are:
The red bridge was known as Guzei and it was supposed to represent the route blessed to take to salvation.
If you look at the Gardening Articles part of Helpful Gardener, there is an [url=https://www.helpfulgardener.com/japanese/2003/zen.html]article about Zen Garden Aesthetics[/url] that quotes this:
Since the Dewy Path
Is that way that lies outside
This most impure world
Shall we not on entering it
Cleanse our hearts of earthly mire? Rikyu
The bridge, particularly the bridge over water, represents crossing over from one world to the next, from the profane to the sacred, cleansing yourself to enter the pure world of nature.
In Zen symbolism red represents life force, preservation, the sacred, and red transforms the delusion of attachment into the wisdom of discernment. https://www.religionfacts.com/buddhism/symbols/color.htm
Thus the symbolism of the bridge is reinforced by it being red, i.e. sacred and transforming worldliness into wisdom.
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