I'm not sure if it is that circular, but it does come up a lot. There are two separate things here though which often are falsely conflated.
1. Having a paid off house
2. Making multiple early payments to pay off the house over the course of many years, short of paying it off completely.
#2 is just a terrible idea. First, as has been noted, there is a 95% chance it will end up with less money.
https://dqydj.com/sp-500-historical-return-calculator/ The thing with 95% odds is that if you do them enough, you usually come out far, far ahead. The odds of investing being better than paying a mortgage are similar to a roulette wheel landing on any color other than green. I'll take both red and black. But the thing that is generally missed (but is probably most important) is the lack of financial security which results from making regular extra principal payments. If you have a $300k mortgage, and pay $200k extra, and then wake up one day needing $100k, you are SOL. That money is sunk. Lost a job? You still need to make house payments, but you don't have any money. HELOCs are more likely to be withdrawn during big recessions when you are also more likely to lose your job. The only way which is at all reasonable is to save that money in a moderately aggressive investment account, and lump it all into the mortgage when you have enough to pay it off all at once. This is both the higher return and lower risk way.
#1 is actually reasonable in many situations. Mortgage rates are higher than most bond yields. Paying off a mortgage while approaching retirement is a good hedge against sequence of returns risk. Saving money in mostly stocks, and selling them when you have enough, increases your odds of selling high, which is always a good thing. I do need to stop and point out that EE savings bonds pay 3.5% after 20 years, so mathematically those are a guaranteed win against a mortgage under 3.5% after taxes. I also need to point out that inflation is running like 5-6% recently, so a 2.625% mortgage is actually a
-2.5% mortgage the past little while. If you want to butter your bread on the same side as the Fed and Treasury (who make money), you might consider holding on to the mortgage.
Also, I'd invite you to look at the every handy investment order thread:https://forum.mrmoneymustache.com/investor-alley/investment-order/
As for what I'd do, I'd save up the money in 10-40% bonds, and a mix of US and international stock funds, or at least enough in stocks to be able to out-run the mortgage with some reliability. Then, when I had enough money to pay off the mortgage, pay the taxes on my capital gains, and still have a little bit of an emergency fund left over, I'd pay off the mortgage in a lump sum if I still felt like it.