@mcmarasco ISTM you're overthinking this.
The 30,000 foot view: What your are considering (investing the $150K) is a form of leveraged investment. It's as if you started with nothing in your pocket and your home paid off, and then you borrowed at 4% (the mortgage) to invest the borrowed $150K at 7%.
That's a net gain of 3% (less after taxes), but as with most leveraging, increased risk. (This also addresses
@davidrmoran's question: what happens if you reduce the assumed rate of return.)
A little more detail: You want to compare two outcomes. The inputs are the same either way are: $150K cash and a 10 year cash flow of $1500/mo. Either way, at the end of 10
years, you'll have your home free and clear. So the only difference between the two paths you suggested is the value of your investment at the end of 10
years.
Path 1: Invest the $150K
@7% rate of return. As you noted, you'll have $300K at the end. (You'll also have paid $180K over the ten
years to reduce your mortgage debt by $150K, so you'll have paid in $30K in interest.)
Path 2: Invest $1500/mo
@7% rate of return. At the end of 10
years, with incremental investments, you'll have about $258K. A lesser result.
Taxes are where one gets into the weeds:
Path 1: The net income of $150K will presumably be taxed at cap gains rate. But you'll also be able to deduct the $30K in interest against ordinary income. We'll assume a 22% rate here.
Your net taxes will be around 15% x $150K - 22% x $30K = $22.5K - $6.6K, or about $16K.
Your total after tax value will be around $300K - $16K = $284K.
Path 2: Your net income is $258K - $180K (the cash flow it cost you) = $78K. Again assuming this is all cap gains, the tax is 15% x $78K or about $12K.
Your total after tax value will be around $258K - $12K = $246k.
[The $258K result came from using
a calculator and investing $1500/mo at 7% annual compounding for ten
years.]