@Pat Jackson you were right! This topic IS divisive! :)
I think someone here said it best... it really depends on your ultimate goal. Here are some things to think about and/or consider....
The most amount of interest you will pay a lender is upfront. The later part of the loan is mostly principal payment anyway. So, paying off a loan that is 2/3 into it's maturity is not a good use of that money as the lender has already collected their portion of the interest in the first 2/3 of the loan term. If you need a visual, go to any amortization table and see how each month the interest and principal is allocated, and how it shifts after year 5, 10, 20....and so forth. So if you are trying to pay as little to the lender as possible, don't get a loan from one. :)
You touched on a great resource at your fingertips...a HELOC. These are great as they can stay open and be used over and over again without incurring closing cost fees associated with originating a new loan. So, each time you pay it off and dip back in, you are just using it like your own bank account. Awesome.
If you have the kind of money that would be enough for a local property at trustee sale, this would be a good way to MAXIMIZE your dollar as you would be afforded the opportunity to purchase a property at a deep discount, and either rehab and flip it, rent and hold it in your portfolio, or wholesale it as-is. If you are selling it, you don't need a loan because you used your cash for a short term. If you decide to hold it, you can refinance it and avoid paying mortgage insurance (giving you more favorable terms and rate) because you acquired it at such a deep discount.
Keep in mind, properties without interest payments don't provide tax deductions....just FYI. You may want to seek counsel with your accountant about what the loss of the deduction looks like versus the added cash flow. It may not even be an issue.
Also as someone mentioned, free and clear properties provide good opportunities of leverage when the market shifts.
I don't think either strategy is bad. I think it all depends on your financial situation, your ability to access funds for your new purchases, and your ultimate goal for investing.