No. Do not upload your music library to iCloud. It is not used for that, and will not in any way make it accessible to the Music App on your new Mac.
If your iTunes Library was in a portable state with all music stored in the iTunes Media folder and the itunes library file pointing to it, you should be able to just hold the Option key on your new Mac when opening the Music app and point it to the iTunes Library.itl file on the external drive.
click here ➜ Move your iTunes library to another computer - Apple Support
Otherwise, you might have needed to consolidate your iTunes Library first.
click here ➜ Make a split library portable - Apple Community