What is the justification for it being a problem if loans are not paid back? As far as I could understand, the loan system applied by banks which is based on a fractional reserve system works as follows:

Assumptions:


The bank has initially 1000$. 
The fractional reserve is currently 10%. 
For simplicity, we assume that all deposits are made only on this bank, instead of having multiple banks. It has been argued in many places that, for the overall view of the money system, that this is a justfied assumption.


The following happens now:

1) Guy A makes a loan for 900\$. The bank keeps 100\$ (more than 10%, thus ok).

Total loans owned by bank: 900$
Total deposits owned by bank: 100$


2) Guy A pays guy A', which deposits the money on the bank again.

Total loans owned by bank: 900$
Total deposits owned by bank: 1000$


3) Guy B makes a loan for 800\$. The bank keeps 100$ as a new reserve (more than 10%, thus ok).

Total loans owned by bank: 1700$
Total deposits owned by bank: 200$


4) Guy B pays guy B', which deposits the money on the bank again.

Total loans owned by bank: 1700$
Total deposits owned by bank: 1000$




This repeats itself a couple of times, until we arrive at:

Total loans owned by bank: 5000$
Total deposits owned by bank: 1000$


From the deposits, 500\$ are reserves of the bank, which belong to the bank, and 500\$ are money which is owned by the last guy that got money from another guy.



Ok, so this is, as far as I understood, the beginning of the "loan process with a fractional reserve system", with a couple of loans. The bank has still enough reserves, so it did everything ok.



Now let's assume that Guy A comes to the bank and says "Hi, I'm sorry, I can't give you the money back, I'm broke".



Currently, the following thing happens: The bank says "ok, so you must give us your security to cover our loss" (for example the car).



Question: Why does this happen, what is the justification for taking the car from Guy A? Since the bank never lost anything...?
It appears that the OP confuses  money with property rights  

A) It writes "deposits owned by the bank" which is simply wrong, since deposits are liquid assets belonging to the persons that deposited them. The bank leases the funds from them and then sublets them to the debtors.
B) Looking also at an OP's  comment to another answer, indeed the bank may create the money, but it creates the money as a medium of exchange and representative of value -it does not keep the property rights on the money itself it creates.
 C) Finally we should not forget that the bank is a distinct legal entity from its own shareholders.

Stopping at the four explicit amounts the OP gave (after person $B$ pays person $B'$ and person $B'$ deposits the money in the bank), the bank's balance sheet is as follows:

\begin{array} {| r | r | r |}
  \hline 
\hline                       
  \text {ASSETS} &amp; \text {USD} &amp; \text{LIABILITIES} &amp; \text{USD}\\
  \hline                      
\text{Loans} &amp; 1,700 &amp; \text{Shareholders Capital} &amp; 1,000\\
\text{Cash} &amp; 1,000 &amp; \text {Deposits} &amp; 1,700\\
\hline
\text {TOTAL ASSETS} &amp; 2,700 &amp; \text{TOTAL LIABILITIES} &amp; 2,700\\
  \hline  
\end{array}

A Balance Sheet is just an inventory at a specific point in time, of Cash and Rights to Collect in the future ("Assets") and Obligations to Pay (also in the future) ("Liabilities"). 

It is important to understand the "Liabilities" (obligations to pay) side.
The initial USD $1,000$ with which the bank started, represent an "obligation to pay" for the bank towards its own shareholders. Second the bank is obliged to pay at any time USD $1,700$ total to persons $A'$ and $B'$ that deposited their wealth to the bank. The fact that the depositing transaction could happen because the bank in a previous instance has created the money, does not give the bank a property right on that money, just because it was its "creator".

That the bank has at the moment $1,000$ in Cash as when it started doing business, relates to the bank's "liquidity" situation. This is not unimportant, and in fact it is an index of financial health on its own right. But it has to do only with the short term horizon. For the long-term horizon though, it is property rights that matter. And so it is not the only financial criterion for the economic health of a bank.

So assume that person $A$ comes in and say "I cannot pay my loan of USD $900$". Certainly, in the short-term, this won't affect the liquidity of the bank -it will still have the USD $1,000$ that it now has. But if the bank writes-off the debt, it will do so against the Shareholders capital invested. So after the write-off the balance sheet will be

\begin{array} {| r | r | r |}
  \hline 
\hline                       
  \text {ASSETS} &amp; \text {USD} &amp; \text{LIABILITIES} &amp; \text{USD}\\
  \hline                      
\text{Loans} &amp; 800 &amp; \text{Shareholders Capital} &amp; 100\\
\text{Cash} &amp; 1,000 &amp; \text {Deposits} &amp; 1,700\\
\hline
\text {TOTAL ASSETS} &amp; 1,800 &amp; \text{TOTAL LIABILITIES} &amp; 1,800\\
  \hline  
\end{array}

Assume things with debtor $B$ go smoothly, the bank will eventually (in the long-term) collect his loan, and it will be able to give to persons $A'$ and $B'$ their deposits back... so where is the problem? Well, the problem is that it will give to its shareholders only USD $100$ back instead of the initial USD $1,000$ investment.  

And who might care?  

First, we live in societies built upon the principle of private property rights. The shareholders have a right to defend their property rights, and the bank acting as their representative attempts to defend them by not simply writing-off the debt (but demanding another asset -the car- in its place), even if such a write-off may affect the property rights only of the shareholders, and not of the people that keep their wealth to the bank as deposits.  

But again, does this creates a problem to the economy now? In terms of actual existing liquidity the answer is no. But the Present is affected by what we project for the Future. After the write-off, the future of the shareholders and of the bank looks worse, because their current wealth is less. And this will affect negatively what the bank and the shareholders will and can do in terms of their involvement in the economic activity now.
So to the question "who cares?" the second answer is "the whole economy".