In the US, OSHA generally requires written workplace safety programs and recordkeeping for companies with more than 10 employees.
Big 'Fortune 500' companies often tout exemplary safety ratings and/or are 'self-insured', or have stringent qualifications placed on them by outside insurers.
The last thing they want to deal with is a subcontractor (like a security integrator) having a lost-time accident or incurring damages and gunking up an otherwise (contrived?) spotless safety records or costing unplanned money in insurance claims.
This is why you see such requirements in a bid response. Sure, pick the best solution for the job, but also pick a subcontractor who isn't a danger to themselves or others while executing it.
Fundamentals of wearing the correct PPE, ladder safety, Lock-Out/Tag-Out, bloodborne pathogens, and construction worksite safety should be core training for any integrator or installer, trunkslammers included.