You're right, if nothing else is wrong and with a good charge in the battery it should start even without the alternator. If the belt has been slipping I'd expect you'd have been hearing some screaching belt noises especially when the engine is cold. Without the alternator, after the belt broke the poor battery got drained and the spark through the coil got weaker and weaker. The engine would indeed start to run weak, stumble and eventually stall and fail to restart. If the battery was old you may well now have a totally dead/shorted battery and even with the alternator reconnected there may not be enough of a spark to start and run the engine. You really need to do some charging circuit testing with a multimeter. After fully recharging the battery, do a basic static battery voltage check. Immediately after a full charge and again after 12 hours or so, it should be able to charge and hold just over 12 volts when not connected. If it still won't restart then further diagnosis is needed -check the OBD connector for fault codes (see the FAQ). Once you get the engine running, check the output voltage of the alternator (again see the FAQ for more info). Have the battery properly tested at the first sign of problems.

Without the fan, the engine would indeed overheat, but with a little luck no serious damage was done unless abnormal steam is/was coming out the tailpipe or the oil looks abnormal (like grey with water). You may want to check cylinder compression in case there was any head damage -even compression say within 6 psi of each other and above 118 psi is what you'd want to see. Obviously refill the rad if needed. Once the engine is running, if the engine temperature does not rise properly then the thermostat may have been damaged by overheating and is simply in need of replacement.

Now if your subject line is correct!!! (and you don't really indicate you know that to be true) and the harmonic balancer (crank pulley) has indeed failed and is internally slipping then you will also have problems from retarded valve and ignition timing causing very sluggish operation leading to rough running and stalling/starting problems. For the B234F 16-valve (an interference engine) you'd want to replace that *immediately* and make sure the timing belt is properly re-aligned. It would be fatal if the harmonic balancer decided to really let go during hard acceleration allowing a crank position where the pistons and valves could collide and do some serious top end damage. A simple test is to physically set #1 cyl at TDC (rotate the crank and use a rod through the spark plug hole) and make sure the 0 degree timing mark on the front timing belt cover still aligns with the scribed mark across the outer edge of the crank pulley. See the FAQ for more information.
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Dave -not to be confused with a real expert