Should there be any issue copying "Library" folders to an external hard drive?

Something is not right my friend....

In Disk Utility>View, select Show all Devices, highlight the top left entry.

Run Disk First Aid on all items in the left panel, from top down.

Are you running any VPN, Anti-Virus, or Cleaning apps?

We need to see what all is running, a report from this will not display any personal info...

Using EtreCheck - Apple Community

EtreCheck is a FREE simple little diagnostic tool to display the important details of your system configuration and allow you to copy that information to the Clipboard. It is meant to be used with Apple Support Communities to help people help you with your Mac. It will not display any personal info. Give it Full Disk Access.

https://www.etrecheck.com/

Give etrecheck Full Disk Access before running.

Thanks for Old Toad’s etrecheck instructions…

Slow iMac 2017 - Apple Community

Which "Library" folders do you mean?

There are several actual "Library" folders necessary for macOS to function which should not be relocated.

Then there are the "Libraries", not exactly folders, which some macOS apps create such as for Photos, Music/iTunes, Video, and Mail. Apple has instructions for relocating these app Libraries to external media....all except for the Mail app Library.

Move your Photos library to save space on your Mac - Apple Support

Change where your music files are stored on Mac - Apple Support

Move your iMovie for Mac library - Apple Support

Of course if your external drive cannot remain connected or reconnect after booting & waking up, then that is a huge problem which you need to resolve before moving anything to the external drive.

You should not copy any folders or files that are in /Library or any user/Library. The files/folders will be useless in any other location except perhaps in certain limited cases for backup purposes.

No and not a wise move at all unless you are fully competent in Mac OS. Based on your question, I think you have more to learn and need to thorougly research before trying to modify Mac OS.