I'm assuming you were having similar issues trying to boot normally to the internal drive?
Did anything recently change before this issue occurred?
First thing to come to mind is a failing hard drive which if the failure is severe enough can also prevent booting successfully from external media. However, a memory issue could also be a possibility.
Try running the online Apple Diagnostics assuming macOS 10.12.4+ has been installed at some point in the past, otherwise you would need to boot the Apple Hardware Test from DVD #2 which originally shipped with the 2010 laptop from the factory.
You can also try creating & booting a Knoppix v8.6.1 Linux USB stick since it should not try to mount the internal drive like macOS or the macOS installer tries to do. If Knoppix boots, then most likely the hard drive is bad. You can run GSmartControl app and post the complete health report for the internal drive here so I can review it to confirm the drive's health. The GSmartControl app is accessed by clicking the "Start" menu icon on the bottom left of the taskbar and navigating the menus to "System Tools --> GSmartControl". You can use the downloaded Knoppix .iso file as a source for Etcher (Mac, Windows, Linux) to create the bootable Knoppix USB stick. Option Boot the Knoppix USB stick and select the orange icon labeled "EFI". The Mac may appear frozen on the Apple boot picker menu while Knoppix is booting, so make sure to give Knoppix lots of time to finish booting....even more time may be needed if the hard drive failure is severe.
You can also try a quick test of removing the internal hard drive and trying to the USB installer or into Internet Recovery Mode using Command + R, or Command + Option + R.
FYI, you will most likely need to use the macOS USB installer for reinstalling macOS on such an old Mac because there have been lots of recent reports that Apple may have turned off the servers for reinstalling older versions of macOS although it should still be able to boot into the online installer to access the utilities of recovery mode.