I'm in it for the long haul, so if the numbers work today the future doesn't concern me as much, but I would hope home prices and rent continually and gradually increased and hopefully within the next 5-10 years interest rates came down to allow me to either refinance and pull a lot of cash out, or greatly reduce my mortgage.But in an effort to get my wife involved and excited about real estate we've switch gears and are currently looking for an STR along the NC coast.

Probably it will be gradually, as soon as a property increases in cash flow.

If you have a small amount to invest you are better of buying and index fund and adding to it gradually.

Is the assumption of gradual appreciation necessary?

I think $200 per door is ideal, but I don't mind $100 per door in the first year and increasing gradually. 

If you are starting out at an avg of only $300/month with no debt, adding debt, and subtracting that from your 300/m cf now, may not get you anywhere.You may need to take a step back, before you can take a giant leap forward, by selling some/most/all of your current rentals (gradually, but not slowly) to access the $1.3M as cash...then use it to flip in order to double your cash.  

Best to do gradually increase or not renew re-lease if you realistically can at those prices.