Hello everyone,
If a HDD has bad sectors, does this mean the HDD should be thrown out?
If I format the HDD in the ext3 format, does it keep information about which sectors are bad and avoids using them?

Also, which Linux tool can scan a HDD and look for bad sectors?

I have Kubuntu 20.04.
This is for a desktop PC.

Bad sectors happen and the hard drive will have a pool of spare sectors to swap in place of bad ones as they are found. The issue is when you start getting lots of bad sectors and the spare pool runs out.
You can use the smartctl command to test and get a report of the drive. This link has a nice over view: https://linuxconfig.org/how-to-check...using-smartctl

Who handles that pool of spare sectors? Is it done by the internal logic of the HDD or is it done by the OS at the filesystem format level?

The OEM maker of the drive usually has diags to use.

badblocks might be used. There used to be some dos option to verify writes and if bad mark the area and try different. I think linux should do that but I can't find it just now.

https://en.wikipedia.org/wiki/Badblocks

"Modern disk drives will probably not show any defective sectors because they silently remap bad sectors to spare tracks,[5] but running the program with a new drive for several days will test the whole surface, and when reading it afterwards S.M.A.R.T. data will eventually show reallocated sectors."

https://www.tecmint.com/check-linux-...rs-bad-blocks/

Bad blocks is only handled by the drive controller firmware, the OS can just query it for info or ask it to run a test.

A reported bad block/sector on a hdd can be as simple as a single ding on the platter. Or it could mean the hdd is about to self destruct (usually slowly). At best, and depending on importance (but what drive isn't important?), I would be replacing the drive. At the least, I would be making verified backups often, and watching for errors indicating imminent failure.

Operating system management of bad sectors hasn't been appropriate for years, decades in fact - it is all handled automatically by the drives these days.

As mentioned you can use smartctl to look at the reallocated sector count. A few, especially on older drives, might be nothing to worry about as long as the number is not rising on a daily or weekly basis. If it is the drive is failing and will die probably sooner rather than later so get any important data off of it before it fails.

mke2fs does still have the "-c" argument to check sectors can be read but with the huge capacities of modern drives it's unlikely to be useful in practice.