This defers capital gains and depreciation recapture taxes on the sale of your fourplex.After converting it into your primary residence, you may qualify for the primary residence capital gains exclusion ($250K single/$500K married), but only for appreciation after the conversion; gains from the rental period remain taxable.

As tenants, I'm wondering what kind of offer we could present to our landlord to buy the house we are renting.

My wife and I were both present when he said he would fix the door for no charge, so it’s not like there was a misunderstanding.

You state there's a lot of deferred maintenance, then state it is a remarkably well maintained property.

Interest income is interest income, unless you decided to roll some of the current cash into a self directed IRA at which point your interest income would be deferred for a later time and allow your funds to grow tax free.

You do have to purchase at least as much real estate as your net sale if you want to defer all tax.TICS are generally cash deals. 

-OOS investors in HCOL areas will buy properties that look good on paper without doing enough due diligence and then get crushed by capex and deferred maintenance.

And you would get to indefinitely defer all tax on the gain from the sale.

So, How does one leverage this very important area of an investors responsibility and (for lack of better way to present it), market or signal to other investors who may be looking to partner?  

Present a win-win scenario.