I agree with @John Galt's astonishment that any reputable tech would just replace the battery without being 100% certain the battery was the source of the problem. The tech could easily have disconnected/removed the battery to verify the laptop was functioning properly....this is possible with those older Macs, but not the USB-C Macs.
A bad HD is a real possibility, but so is a GPU failure which was really common with the 15" & 17" models of the 2008-2011 models. Bad memory or a bad memory slot is also another possibility. There is no way to know without proper testing & troubleshooting. A reputable tech should have no problems performing these simple checks.
You can even try running the Apple Hardware Test which you should be able to access by booting from the 2nd DVD which originally shipped with your laptop from the factory. I forget if you need to hold down "C" or "D" to boot Apple Hardware Test from that 2nd DVD (I think it requires holding the "D" key). Unfortunately the diagnostics are unlikely to detect a bad GPU, and it may not even detected a failing Hard Drive unless that drive failure is severe enough. With a third party battery installed, the diagnostic may report that third party battery as being bad even if it is perfectly healthy. The only useful diagnostic is one which reports a failure (assuming all original OEM components).
You can try installing macOS onto an external USB drive, but keep in mind installing to and booting from a USB drive on this old laptop will be extremely slow due to the very slow USB2 port, however, it is good enough for testing the laptop. If booting from an external drive works fine, then most likely the issue is with the internal boot drive (software issue, file system issues, or even drive failure), however, if you have trouble installing or booting from the external boot drive, then it may point to a wider system issue or possibly a Hard Drive with a very severe failure.
Creating a bootable MemTest86 USB stick & booting from it is also a simple way to partially test the system. While not as informative as using a full macOS external boot drive, it may provide a bit of insight. Creating & booting from a Knoppix Linux USB stick is an option as well since it is a full non-Apple OS running entirely from memory once it boots. However, the system may appear frozen on the Apple boot picker menu while Knoppix is booting so you need to allow Knoppix some time to finish booting....this was one of my favorite ways for testing older systems.
FYI, here is an Apple article for various boot issues....I'm not sure if these articles are still completely accurate for such an old system (I doubt it mentions using your OSX 10.6 Snow Leopard Upgrade DVD to get started with reinstalling macOS so you can install macOS 10.11 El Capitan)....you may need to locate the comparable articles on the Internet Archive's Wayback Machine.
If your Mac doesn't start up all the way - Apple Support
Edit: Booting from a full macOS installation on an external drive or from a bootable Knoppix Linux USB stick will allow you to actually check the health of the internal drive.