If Snow Leopard and Yosemite are booting and working fine, then it should be possible to install macOS 10.11 as well without erasing the drive. However, if there is a problem with the file system it may be interfering with the 10.11 install process. By erasing the whole drive (possibly re-partitioning it) you eliminate the file system issues unless it is caused by a hardware issue. I forgot to suggest to run Disk Utility First Aid on the drive and to run "Repair Permissions" as well (I think the "Repair Permissions" may still be available in macOS 10.11, but if not then run it from Yosemite).
I don't see anything in the DriveDx report although the short self-diagnostic never finished which is unusual. Many times I find that software issues will cause this problem because it keeps the drive too busy in order to successfully complete the selftest. Since the short diagnostic usually only needs to two minutes (maybe 5 - 10 minutes if the drive is being used heavily), this could actually indicate a hardware issue either with the drive, the hard drive SATA cable, or possibly some other issue with the laptop itself. See if you can get the short self-diagnostic to complete while booted into Safe Mode.
It is possible to run the long/extended self-diagnostic using the Terminal app & command line even during the trial mode of DriveDx by using this as a template:
sudo <path-to>/DriveDx.app/Contents/Resources/smartctl -t long /dev/disk0
You can run the long/extended self-diagnostic by typing "sudo " in the Terminal making sure to leave at least one space at the end of "sudo ". Then drag & drop the DriveDx app onto the Terminal window using the Finder to auto fill the correct path to the DriveDx app. Then in the Terminal app press the "Delete/Backspace" key to remove the trailing space and so the cursor is up against the letter "p". Then continue typing by adding the following:
/Contents/Resources/smartctl -t long /dev/disk0
If the DriveDx is still located in the "Downloads" folder, then the command would look like this:
sudo /Users/hwtech/Downloads/DriveDx.app/Contents/Resources/smartctl -t long /dev/disk0
Once the command is entered correctly press the "Return" key to execute the command. You should be able to review the results using the DriveDx app itself. The long/extended self-diagnostic will give you an estimated time for completion shown in the Terminal app which may be as long as an hour. However, since the short test is not completing in a timely manner, then the long/extended test may take much longer (even a day). It would be best to run the test while booted into Safe Mode.
As for your macOS 10.11 installer. Either you have some sort of software issue which is preventing the DriveDx short self-diagnostic to complete and interfering with the installer when it is run from within macOS or you have a hardware issue.
For the bootable macOS USB installer use the official Apple instructions found in this Apple article to create the macOS USB installer:
https://support.apple.com/en-us/HT201372
Apple recently changed the macOS 10.11 installer and it is possible Disk Maker X has some type of compatibility issue with the new style 10.11 installer. I know TransMac users have reported issues as well.
If you have issues when booting/installing from the USB installer when following Apple's instructions, then you may want to try using another brand of USB stick or use a USB hard drive/SSD for the installer instead. Macs can be quite picky about the USB sticks used for booting and the quality of USB stick is quite poor.
Another option would be to try and install macOS 10.11 to an external USB or Firewire drive to see what happens. This should work even if the system will run extremely slow while booted to the external drive. It at least will verify your USB installer is good and most of the laptop's hardware is working.
