Looking for a good application to remove old files

There is one simple and safe way (where you decide)

I would not trust any Automated app deleting stuff that I might want to keep.

1. Click on the Apple Icon (top left)
2. About this Mac
3. Storage
4. Manage
5. Reduce Clutter

Download and run Etrecheck.  Copy and paste the results into your reply. Etrecheck is a diagnostic tool that was developed by one of the most respected users here in the ASC and recommended by Apple Support  to provide a snapshot of the system and help identify the more obvious culprits that can adversely affect a Mac's performance.

Use the Add Text button to include the report in your reply:

Also assign Full Disk Access to Etrecheck so that it can get additional information from the Console and log files for the report:

There is not such an app - how would an app know what you consider "old"? If you want to delete old applications, check into it and/or their developer to see if they have uninstallers. If so, follow those directions. If not, they were likely installed by simply dragging them into the Applications folder and it is safe to drag them to the trash. Don't worry about any small files they may leave behind.

None of that will actually "speed up" your Mac. Keeping enough/plenty of hard drive space and having enough RAM will be the best strategy along with this caution: don't try to have too many apps/processes running at the same time; you have an older processor so you need to help it by not doing too much at the same time.

And, remember, it is 10 years old, that is considered ancient for an electronic device.

The report shown none of the usual suspects.

I agree with dialabrain on the erase and install recommendation. That would be easiest and you could only install those apps that you want.

I used the wrong image. Try this button:

FWIW, the easiest thing to do would be to do a clean install. In other words erase the drive then reinstall the OS. Then install any extra apps you want to use. Create a clean slate.

The solved should have gone to dialabrain as he suggested it initially. I only seconded his recommendation.