This is incorrect. Bank solvency has to do with the assets of the bank, even in markets with commodity (e.g., gold) or representative money (e.g., gold-backed paper). The assets of a bank include loans, the liabilities are the money the bank owes to depositors. There is no reason a bank can't take a gold deposit and loan it out (thereby, "creating" gold).
The purpose of a central bank, at least in orthodox economics, is to loan to solvent banks that are nonetheless cash-poor. Imagine a mismanaged bank, that has loaned too much, and cannot meet the demands of depositors. If the loans + cash are more valuable than the deposits, the central bank will loan to the bank to meet their temporary cash shortage. [0]
This is essentially what happened in the case of AIG. The Fed believed that the value of AIGs assets were greater than its liabilities, and loaned them the money at a penalty rate. The Fed believed that AIG was _solvent_. There were a number of extenuating factors here that I'm glossing over, but that is the underlying point. The reason that Lehman was not saved was that the Fed had substantial reason to believe that the assets (primarily the sub-prime loans) were not worth more than the liabilities, and the Fed will not lend into hole. Lehman was _insolvent_.
Fractional reserve banking, by itself, does not suggest solvency or insolvency. Without fractional reserve banking, there cannot be credit. Sharia banking is an example of full-reserve banking, because interest is prohibited, so there is no incentive to loan. (There are ways Islamic banks get around these prohibitions).
[0] I would suggest looking at Bagehot (1873) for a full description of this idea.