Welcome to stack exchange. This site is usually for DIY residential owners, because when you get involved in commercial you usually will need a professional electrical contractor, but I will take a crack at your question.
First your ground attached to your first means of disconnect is the building ground. It can't be a 6/0 since there is no such gauge. It is probably a 3/0 or 4/0 depending on how it was originally sized. All other grounds from your tenant panels must be sized and connected to this ground. So yes all of your tenant services need a grounding conductor to the original ground.
Second on commercial properties, if you panel grounds are mixed with your neutrals they need to be separated and installed on proper buses. Neutrals need to be on a neutral bus and grounds need to be on the ground bus, and if the panel is the first means of disconnect the neutral bus must be bonded. If you have a separate main breaker or fused disconnect the neutral must be bonded there and the neutral bus in the panel and the ground bus must be isolated from each other.
Third NEC Article 250.50 states that all grounding methods available shall be bonded together to form a grounding electrode system. I can't say how many are available on your building but there should be at least two. One is the building steel you described and two would be a driven ground rod.
Conclusion - If what you described is factual; then no your grounding system is incorrect and needs to be repaired and brought up to code. I might add that proper grounding is important not only for safety purposes, but for the proper operation of your electrical system. I can almost grantee that your tenants will have problems with there electrical and electronic equipment, and if you call in someone to trouble shoot it. It will take longer to trace out and cost more to repair.
Still I recommend that you bring in a recommended and trust worthy commercial electrical contractor to get you an "eyes on" inspection and recommendation.