How much cash do you currently have invested in the deal? There is something to be said for every strategy, but the significant advantage that we have in real estate is leverage. You could approach this in a more hybrid manner, one that would make you feel better about saving money on the interest and snowballing into more properties quickly. There is no need to wait for the first property to be paid off before buying the next property, and putting some extra money toward the principal is an excellent option to reduce that interest payment. The thing to be careful of is optimizing your capital and leverage.
For example, if you pay off your property, what is your ROI on that cash invested? I'm going to assume your property is worth around $110,000 (based on what you are saying), so it fully paid off, making $700-$900 cash flow. Your ROI is 8.7%. Now, on the same property with the leverage you have, assuming 34,000 in the deal currently and $250 cashflow, your ROI is 8.8%, so the same after the cost of debt.
DISCLAIMER: This is not a super in-depth evaluation of the numbers. I am using a lot of rough numbers and generalities but trying to demonstrate a point.
Option 1 Leverage: We will assume you plan to hold all properties for the next 30 years. Lets assume that you are buying similar properties to the one you have now. I'm going to assume you are able to save up downpayment money for a new property every 24 months, which would leave you with 15 properties over the next 30 years. Now, let's assume a modest 3% appreciation over that period. Your first asset would be worth $267,000, and your total portfolio balance would likely be around $2.5+ Million. Your first house would be paid off, and every two years after, another house would be paid off. You would have needed to bring approx. $500,000 as a down payment, and considering an ROI of 8.8%, you would be bringing in $44,000 per year, and that would grow exponentially as you paid off each house. Your appreciation for year 31 would be on the $2.5 Million, so around $75,000. (Total Portfolio Value: $2,575,000 | Portfolio Debt: $1,470,000 | Portfolio Net Worth $1,030,000) Honestly, the returns are even better than this when you take into account tax benefits, but I'm trying to simplify for an example.
Option 2 Pay Off: In this example, you buy a new property every 4 years after the first is paid off. It's the same 30-year time frame, so you would end up with 7 paid-off properties at the end of 30 years. Your total portfolio would be valued at around $1 Million. Now, assuming you are at the same 8.7% ROI on that you would be bringing in $87,000 in cashflow and your year 31 appreciation would be $30,000. (Total Portfolio Value: $1,030,000 | Portfolio Debt: $0 | Portfolio net worth $1,030,000)
Conclusion: your ROI looks about the same at year 31 one option has a higher portfolio value and is leveraged. The other is cash flow heavy with a lower total value. Now, let's look at year 40 (remember 3% appreciation).
Option 1: Total Portfolio Value $1,384,000 | Portfolio Debt: $0 | Portfolio net worth $1,384,000
Option 2: Total Portfolio Value $3,359,000 | Portfolio Debt: $550,000 | Portfolio net worth $2,809,000
Ok, this is a super long post with some rough numbers, But I'm mostly trying to demonstrate that leverage helps snowball faster, and the advantage of appreciation on a larger total portfolio balance can be a huge tailwind when investing in real estate. You mentioned that you are less concerned with cash flow right now, and that signals to me that you are probably better off finding the sweet spot for you with leverage.
I would leverage and take on extra risk now while you are young and able to make great money, and maybe 20 years down the line, switch things up and start pushing to pay these properties off early. You could also build up an emergency fund for each property, and then once you have plenty set aside for repairs, put all your extra cash flow toward the principal and pay the properties off faster. For your first property, if you put your cash flow of $250 a month toward principal payments each month, you would pay off 16 years early without hindering your savings effort for more properties.