This is a long thread and I only read the first page or two of responses. However, I wanted to add another resource to the discussion. I skimmed the rest of the thread to see if anyone else already mentioned this explicitly but does not appear so. Forgive me if someone has spelled this out already and I overlooked it.
Keith Weinhold talks about this conundrum a lot at Get Rich Education. He provides his perspective occasionally on his podcast (and has an episode where he talks about it in depth), plus he apparently practices what he preaches by minimizing equity, even in his personal residence (he says). Disclaimer: Keith does fall more towards "guru" on the spectrum and I wager he probably makes some good money off the referrals, etc. that are found throughout Get Rich Education. But that said, I think Keith does provide some really good information and conducts interviews that are worth listening to, so please look past the rest.
His views are summarized at https://www.getricheducation.com/why-home-equity-is-an-awful-investment/. To briefly outline, he advocates minimizing home equity because it's:
1. Unsafe: If your house loses value, the first thing it impacts is any equity you have in it. The hit to the bank comes afterwards, leaving the bank in a nice position with a decent hedge against value decline: your equity. So if you pay 20% towards a property and the value goes down significantly, it is just like losing that money. Of course, if you are holding for a longer period and can get past these dips, you are likely fine, but if forced to sell, you get nothing and the bank likely gets most of what it wants. Why not pass more of the risk to the bank by minimizing equity?
2. Illiquid: If you have equity beyond ~20% then there is some degree of liquidity, but it costs a fair amount of money, time, and effort to access that equity through a sale or refinance. If you have < 20% equity, in many cases you have essentially no access to your money unless you sell the asset, not to mention the added cost of PMI. I don't think anyone would dispute this.
3. Not earning anything: Whether you own free and clear or have a mortgage, the amount of rent you can charge on an investment property is the same. So having that equity gives you nothing more than having the property totally financed. Of course, the amount of financing you have will impact the cash flow after all expenses, but ideally, we are all buying right and ensuring we have decent cash flow. With leverage, ROI is likely much higher. This argument does not hold quite as much water for me as the other two, but he does have a point and in the interest in full disclosure, I'm providing this argument also.
I think the key to minimizing equity is ensuring that you have emergency funds or capital available if you lose a tenant, have a disaster, etc., which you should have to some extent anyway. I guess Keith's argument might boil down to the mantra: minimize equity and manage risk through adequate reserves/insurance. Perhaps you have the same amount invested in low equity and good reserves as you have in a traditional 20% down mortgage with minimal reserves. But with the former, at least those reserves are liquid, within your control, etc.
In my opinion, this is a strong argument for minimizing equity, but perhaps I am relatively risk-tolerant and willing to use leverage. It may not be for everyone, but I thought I would spell it out here for those who are learning or want to see another perspective.