Any Inspection Template created on Company Level is disabled on project level by default.
Don't forget to also enable the inspection template on project level so other users can view and add it.
Website
Login to a project.
Go to Project Administration > Inspections.
Check the box at Enabled? Column and click Save.
If the inspection template is enabled and you cannot add the inspection still, you may want to check the form permissions at User / Contractor accounts. Learn more here.