In the event of a run, a bank is considered illiquid, an insolvent bank is one where losses on debts exceed loss provisions and capital.
An entity that only wrote loans, and didn't have deposits would not be a bank - the definition of a bank is implicitly that it is performing fractional reserve banking via double entry book keeping. (Unless it's the World Bank, which is actually a fund, because the US and UK had an argument about who would control the International Monetary Fund (which is actually a bank) when the Bretton Woods agreement was setup.
No banks, including central banks can really be described as robust. They have at best around 1% fault tolerance in terms of the quantity of loans as a percentage of total lending that they can write-off each year.
Yes, that is about to become a huge problem.
https://www.federalreserve.gov/newsevents/pressreleases/mone...
This indicates that the traditional story of how fractional reserve banking causes money growth is wrong.
In reality, even a non-zero reserve ratio doesn't limit money creation if you look at the financial system as a whole, because the created money will simply become deposits elsewhere.
The true limiter of money creation is capital constraints: somebody must give money to the bank and be willing to lose it - only then is the bank allowed to make loans.
The two issue's with non-zero reserve regulation were (historically) the price of gold was implicitly linked to deposit expansion, and in the Bretton Woods era, all the countries had different expansion rates. Then post Bretton Woods, Basel comes in.
With Basel the Bank's risk weighted capital also regulates the amount of lending, and hence deposit expansion.