If the battery failed a load test and the load tester was set for the correct battery CCA (cold cranking amps) then the battery is not any good, and if you can dis-connect the battery and using a volt meter you can watch the battery voltage drop off until it is below 12 volts, then the battery is once again not any good and that battery most likely has an internal short or a bad cell and that alone would cause similar symptoms.
The alternator could cause a draw on the battery, but the starter will not unless it is engaged all of the time, and if the mechanic at the service station that you went to does not know that, then you might want to consider another repair facility.
There might be another draw on the battery and the best way to find it would be to dis-connect the negative side battery cable and connect a test light between the negative battery cable terminal and the negative battery post. (see image)
With everything turned off and the doors closed, connect the test light between the negative battery cable terminal and the negative battery post, and the test light should light up bright for a few seconds and then dim down very low to almost nothing indicating that only the KAM (keep alive memory) for any control modules and the stereo system is active and drawing power from the battery and this is a normal condition.
If the test light does not dim down to almost nothing, and the test light stays bright or only partially dims down, then there is a draw on the battery from the vehicle electrical system, and the best way to locate it would be to start removing the fuses/relays one at a time (replace the fuse/relay if there is no change in the test light) and observe the test light, and if you do remove a fuse/relay and the test light does dim down to almost nothing, then the draw will be in the circuit that fuse/relay controlled, and if there is still a draw on the battery after removing all of the fuses and relays then start dis-connecting power accessories like the alternator one at a time until the test light does dim down until the test light is just barely making the filament in the test light bulb glow.