I don’t think there’s a way to auto-distribute staves evenly, because variations in the music (particularly high/low notes or stacked notations etc) mean different staves/systems tend to require different amounts of vertical space.
You can set a fixed casting off value, e.g. if you have 20 systems total, set this to 10 systems per frame. You can also handle this manually by inserting a frame break at the start of the system you want to be at the top of the 2nd page. Alternatively, you can tweak your vertical spacing settings, but that’s less directly to produce the direct results you want as quickly.
(In general, removing or hiding information in text frames outside the boundaries of music frames won’t cause Dorico to recalculate spacing, as that’s determined within music frames. I know you can’t see those in Elements, but your best bet to managing how music music fits on the page is to look at the vertical spacing settings, casting off, the music frame margins, the page margins, and the staff size.)