How can I change my iCloud plan billing date?

If you're talking about the day a subscription renews, you would have to cancel the subscription, wait for it to end (expire), then start it again on the desired date. Note that for some subscriptions this may entail loss of data (such as playlists for Apple Music) or other features.

View, change, or cancel your subscriptions - Click on the link to learn "How to cancel a subscription from Apple." --> If you want to cancel a subscription from Apple - Apple Support

I can think of several reasons how adding that feature would result in over complexity in the financial dealings since not only is Apple involved but also the developers of the apps. You are welcome to send feedback to Apple (remember, the people posting here do not work for Apple) but since it has been this way for many, many years now I suspect they know that a few people want to do this but it isn't worth the trouble.

So the company that effectively created the smartphone revolution, made digital media mainstream and has changed the way the world talks to itself, cannot change my billing date? Really?