Is the computer a managed device by your employer? I have two MacBook Pros, one an employer device, the other my personal device. Both are set up much the same except the employer one is "managed" which means it can be accessed remotely, configured by employer, etc. Both do Time Machine backups but the employer one often takes notably longer to complete and also has more noticeable disk activity of the sort you mention. Hence my question about managed devices.
I believe indexing shows up as disk activity (especially disk reads) in Activity Monitor, it is reading from the disk, and would be associated with mds processes. However, mds is almost always active in one form or another. Also, I believe you can exclude an external Time Machine backup drive from Spotlight via the Spotlight system settings/preferences. This will not prevent you from restoring from backups.
Disk Utility only reports on the integrity of the file system, not the hardware. You are hearing hardware noises, hence my suggestion to run DriveDX.
Have you tried completely disconnecting from the internet and any local network as well, to see if the noise continues, goes away, changes, etc. Indexing is a local activity and hence would continue if that is what is going on.
I use several mechanical and solid state drives for Time Machine and the mechanical ones are noticeable noisier and have such activity that lasts much longer. You might consider replacing it with an SSD if the noise bothers you.