To answer your question, Yes you can buy an umbrella policy that offers you only General Liability coverage on your properties and will be blanket coverage for all of your properties.

AND with an additional caveat: the PROVABLE cash flow from the property must service the debt with a cushion of at least 20 -25%, so a debt service coverage ratio of 1.2 - 1.25.That being said, the 'gap' can, in some cases, be filled by seller carrying back a second lien. 

The only reasonable explanation I can find is they are supplementing with J.O.B. income, or perhaps living off the excess borrowed funds..Some part of their story is missing.

I'm hoping to earn some extra income to supplement my day job.

Many title companies are run by lawyers who want to stick to title and not have to pay for malpractice coverage.

I am not large enough (16 SFRs) to self insure because I need some degree of coverage for catastrophic events. 

I recommend you have some amount of coverage for catastrophic events and injuries.

I feel we are at a place in the business where I should get this sort of coverage.

If that's not an option, getting a side hustle to supplement income could also work.

Other houses that I have that are strongly cash flowing I can use to supplement my income and help with operations costs when starting into the MF property.I have also invested in a pair of fourplexes, which is not quite the multi-family model (which is 5+ units) but a good starting point with a larger supply of units so a vacancy does not hurt quite as much.