As expensive as batteries are, alternators are even more costly. For that reason, we always recommend that any battery that needs to be jumped, be fully-recharged with a battery charger as soon as possible.
Just as you would likely visit a gas station as soon as possible, if you used a 1-gallon can of gas to get your car started, you should treat jumper cables as the electrical version of that spare can of gas- they can help get you out of trouble, but they may not keep you out of trouble.
Just because a battery is deeply-discharged, doesn’t mean it needs to be replaced either. Many of the “bad” batteries returned to us under warranty are just deeply-discharged and work fine, when properly-recharged.
If you have a good battery that has not been repeatedly discharged then charged over a period of time then yes, it will completely charge it just by jump-starting it and driving it. I have done this many times with my own vehicles and on farm equipment as well.
However, if you do this too many times it will cause the battery to fail prematurely and in that case this method of “charging” would not work.
It is important to note that the alternator is not meant to be a battery charger, it is meant to be a battery maintainer. It is true that it can charge the battery but that is not it’s main purpose.
If the alternator is used for this purpose too many times, then it will also fail prematurely.
Also, it is not likely that your battery is ever “completely dead”. Even when the battery is discharged it can still have upwards of 9 volts. If you put a voltmeter on a battery and you get a reading of 0 volts, then you may very well just need a new battery.
The fact is though that 95% of batteries are never fully discharged when they need a jump start.
A plug in charger is better, but you can just let the alternator charge if it is absolutely necessary.