Assuming the watch doesn't run but the main spring is fully wound (reasonable assumption as you are trying to let the spring down), the fact that the watch runs when additional power is applied through the crown suggests that there is unlikely to be too much amiss with the balance / escapement, rather that there is excess power loss through the gear train sufficient to stop the movement. Even a totally worn out main spring should get the balance moving (albeit with low amplitude) if the train is in good order.
The possible causes of power loss within the train are numerous and include;
1. Dried up / old lubricant or other contaminant fouling the pivots
2. Bent, badly worn, or broken pivots
3. Cracked or chipped pivot jewels
4. Insufficient end shake or misalignment of wheels due to poor workmanship / damaged plates or bridges
This is pretty much the order of frequency in which I encounter power loss issues with degraded lubricant and dirt accounting for probably >80% of the problems.
My first course of action would be to strip and thoroughly clean the movement which will solve 1. in my list.
Then a careful inspection of each part under magnification prior to assembly should identify any issues described in 2. and 3. which could then be remedied.
Then I would reassemble the train all the way from the barrel through to the escape wheel and check for free running. I do exactly as Geo describes in another post, I use a rubber bulb type air blower to gently puff air onto the rim of the escape wheel. If all is well the whole train should move easily all the way back to the barrel, and it should work in both directions. If it doesn't then I strip it back down again and reassemble one wheel at a time, testing at each step for free movement to identify where the problem starts. I can then try and work out what the actual problem is and how to rectify it.
Finally, once I am happy that the train is free running, the click and ratchet wheel go back on and I install the pallet. At this stage, with just a small amount of power on the main spring (less than a quarter of a turn usually), the pallet should snap across between the bankings when nudged with a dry oiler if all is well.
If I get to this stage and all is fine then I would expect the watch to run when the balance is installed. If it doesn't then my focus usually shifts to the balance.
Hope this helps.