Leverage gets you more cash on cash returns. What difference does what the bank is making make to you? You're making more money than you would without the leverage.
The ability to float the payments in case of vacancies and a downturn is a different story. Having a contingency plan and not being stretched too tight as to be wiped out by a couple of bad months is key. Paradoxically, you're likely better off taking a bigger mortgage and putting the extra cash in an emergency fund than you are putting in all in initially and keeping mortgage balance as low as possible.
There's a right balance for everything.
Paying down a mortgage is essentially converting one asset (cash) into another (equity in real estate) while saving the interest cost.
High leverage runs the risk of a wipeout in case the asset value falls. But having the (balance sheet) equity tied into real estate rather than fully liquid and safe, won't always be advantageous.
Take two hypothetical homeowners. A used his cash to prepay his mortgage, while the B built up cash reserves (possibly in a life insurance policy) and just made his mortgage payments as they came due. In 2008 property values went down by 50%. A had a small 125,000 balance left on his mortgage, but no cash war chest, while B still owed 500,000 on his mortgage but also had a liquid war chest of 200,000. B was able to deploy his war chest opportunisticly and buy a neighboring property out of foreclosure, and was also able to modify his own mortgage, slashing his monthly payments by 40%, while A just felt bad for all the people who could lose their homes to a foreclosure (though he knew of none in his community).