You can partially do that already (if I got your idea right) in OpenMPT:

Pressing Insert key on the keyboard while a row is selected will move this and all rows below down by one row.

To do that across multiple channels, create a selection across all channels and again press Insert.

To "move up", instead of pressing Insert, the key is Backspace.

Limitations:

• Moving down below the last row in a pattern will delete the content of it, it doesn't wrap to the next pattern or to the beginning of the same pattern.
• Pressing Backspace on a selection with content will move up everything below this row and delete that row.

While the suggestions are workaround, I feel that the original request in itself is already a workaround: Semantically, there is no good reason why inserting rows should move the last row to the top of the pattern for what you want to accomplish.

When working with a fresh song, having some blank pattern(s) right after the current pattern and using global insert/delete would be semantically more consistent. Otherwise, maybe inserting a row should push the last row of the pattern into some invisbile scratch space at the end of the pattern, from which notes could be recovered by deleting a row again? Though that could also lead to some surprises if you inserted a row 10 minutes ago, then went to work on a different pattern and now delete a row again and suddenly some note shows up at the end of the pattern and you have no idea where it came from.