I'm not defending anything, only suggesting that the foundations of the criticism of fractional reserve banking are very reasonable and rational points.
Your quoted example is oversimplified. If the Rothbard bank gets a deposit of $1000 it must put $100 on reserve and only has $900 to loan. If someone borrows and then deposits all $900, then the bank can put $90 more in reserve and has $810 to loan ... after an infinite number of borrow/redeposit transactions then all $1000 is on reserve and the bank has lent $10000 creating $9000 in money supply (same number you arrive at, but it's NOT in one fell swoop. And yes this is inflationary.
The reality is alluded to in your second paragraph. The guy who borrows the $900 obviously intends to spend it - so he will not keep it on deposit long if at all. He transfers the money to a 2nd party who may well deposit it. ... the $810 to be loaned to a 3rd party and redeposited by a 4th and so one.
The consolidation of all the banks $10000 loan in a single borrower is a case that does not occur. The bank obviously wants and needs and generally a divisified portfolio of loan assets.
Special economic conditions can cause a large fraction of borrower to fail to repay at the same time, or else a large fraction of depositors to want to withdraw at the same time (see the news for examples). In such cases a backing central bank (federal reserve for the US) issues credit to the banks until the crisis subides.
The example presented is not a credible counterargument to the value of fractional reserve banking.
Although the fractional reserve system causes the creation of money supply for additional deposits, and that is inflationary, this inflation only occurs when there is "new money" to deposit. Nearly all of the existing M2 and M3 are already on deposit with associated reserves - these aren't being multiplied further. Also note the central bank or government has complete control over this since they can increase the reserve requirement to reduce the money supply. More commonly the Fed issues or repurchases bonds to adjust the money supply.
The terms "counterfeit" and "pyramid" in your quotation, marks the article as incendiary prattle from some paranoid fool.
Yes it's a complex system to analyse, but not beyond a 1st year book on economics. No it is neither a pyramid scheme or a license to print money.

