A good question, with variable replies
In real terms, if the External Hard Drive (EHD) with Time Machine backups is failing, then that data would be tricky to access never mind copy to another EHD.
And my view would be to start afresh with a new EHD. After all,
TM is not meant as a BU storage facility and if all is well with your Mac now, then starting afresh is the way to go.
My practice is to have 2 TM BUs running alternatively (all modern Mac allow this) so that if one fails, I replace it whilst keeping the other in case of need.
Another answer to you question could be that if you Cloned the EHD with the TM to a new EHD, that might work. I've never tried this and I'm not sure that the exact integrity of the TM backup would be maintained, although technically a Clone is an "exact" copy. You would need to use an app such as Carbon Copy Cloner (CCC) or SuperDuper! (SD!).
Bottom line: In my view, start afresh. Don't trust the failing Drive. Use 2 TM BUs in the future if you have concerns. (or one TM BU and one of the Cloners, as above)
Ian