If by re-installing you mean a repair install then you may still see the UAC prompt.
http://www.tenforums.com/tutorials/16397-repair-install-windows-10-place-upgrade.html
As you say a clean install will mean you lose your programs and have to find and install them all again.
Did you previously try to enable the built in Administrator account?
You need to access the recovery environment first. If holding down the shift key an selecting Start > Power > Restart doesn't work then open a cmd prompt and type shutdown /R /O -T 00 press return.
Select Troubleshoot and then Command Prompt are you able to enter net user administrator /activate:yes press return. If it is accepted the restart and you should see the Administrator account on the sign-in screen so use that to sign-in and go to Control Panel > Users and see if you can set your normal account as administrator.