It really truly depends on the check and the areas of responsibility.  For example my organisation has a facilities management company who is responsible for all of that so I then wouldn't expect anyone from my team to carry out checks and inspections the FM team are responsible for.

The person completing the checks needs to be competent too - hence I wouldn't go and carry out checks on a forklift as I don't use them and wouldn't know what I'm looking for, neither would I expect the forklift driver to come and carry out inspections and checks on fire doors! Any check should be carried out by someone with the relevant knowledge and experience.

A simple tag system for something like preuser checks on tooling etc would be ok.  It doesn't matter what tool you use, it's the behaviours you need to influence.  You can use the most sophisticated tool going but unless people buy into it and want to use it, or at least understand why they need to use it, then you'll be back at square one.