How to best confirm a subcontractor's insurance certificates and their actual underlying insurance coverage?

When we start a job or hire new subcontractors, it's part of our prequalification process to confirm that they are insured. AND, importantly, to figure out who their subs are (if any) and verify that they are also insured. The standard practice is to collect certificates of insurance (COI). And we do this. However, there are situations when the COI looks adequate and meets limit requirements, but then the underlying policy has exclusions that would limit or eliminate the coverage all-together. For example, some policies limit coverage whenever a contractor is doing work at a certain height (i.e. more than X stories high)...but the contractor is working on a tall building that violates the policy. From the COI, there is really no way to know this, but the insurance policy will offer no coverage. Yet, we collect the COI and mark the subcontractor as insured. They really, of course, are not insured. Requesting actual copies of the insurance policy and all included exclusions is possible, I guess, but what a burden that seems. Getting the full policy is not part of the industry practices and will likely cause a burden to someone (i.e. the subcontractors and/or their agents), but more importantly, it then requires our team to become insurance policy reading experts, and these things can be really complicated.