It's not easy to keep many plants with koi. At least, that's been my experience. I follow some koi forums and they adamantly recommend water changes, as many of their ponds have no plants.
Koi can be fairly destructive to aquatic plantings. Many dedicated Koi keepers have added separate areas for aquatic plants either as a supplemental pond or as part of a wetlands filter. There are those hard liners, however, that want only what amounts to a large outdoor display tank for exhibiting their fish. Yes, these types of ponds will, over time, show elevated Nitrate levels and water changes are one of the ways to combat this. However, simple math shows that unless these water changes are massive (40%+), no real reduction in Nitrate is realized requiring these same water changes to be conducted periodically. These are, without question, stressful to the fish, perhaps even moreso than the perceived danger of elevated Nitrate. (There is no scientific data supporting the claim that a Nitrate level under 100 mg/L is harmful to fish even over extended periods of time. The fish adapt.)
As to the math involved.
Example- A pond has a Nitrate level of 60 mg/l. A 20% water change is performed which means that the Nitrate level is reduced 20% or from 60 mg/L to 48 mg/L. But one must add back in the Nitrate level of the new water which can vary from 2 to 3 mg/l for municipal water to much higher for some well water. So, at best, the Nitrate has been reduced to 50 mg/L. According to present beliefs circulating in the world of Koi Kichi this is still high. If these water changes are scheduled weekly or less frequently, the Nitrate level will have risen back to its original level of 60 mg/L or higher. Changing a higher percentage of water will further reduce the Nitrate level but in the process also drastically changes one the the pond's main water quality parameters. It is an established fact that an abrupt and major change to
any water quality parameter
can or will severely stress not only the fish but every other aquatic organism, but there are those that would have you do this on a regular basis.
Water changes are a holdover from the early days of aquarium keeping when the hobby was in its infancy. Aquarists have since learned the advantages of the 'planted tank' as well as other less intrusive ways of controlling Nitrate levels.
And to those that present the argument that natural ponds and lakes are flow through and experience constant water change/replacement, I would invite them to test the water being supplied to these ponds and lakes. They will quickly find that the Nitrate (and Phosphorus) level in these source waters is almost always higher than the levels in the ponds and lakes that they are feeding.