The whole reason to vote to punish the poacher at all is because not doing so would look soft to the nobility of the realm, and going out of our way to help out the wife and daughter because we were affected by his story is also going to look weak, at which point, why not spare him?
Because a criminal judgment is a balance between multiple interests for someone in our position. If you want the pure political argument:
Being seen to punish poaching harshly is important for our relations with the nobility of the Seven Kingdoms collectively because they take it as a signal that we will look out for their interests and their perceived rights in a political system that both privileges them and greatly empowers them.
Being seen to exercise mercy for the dependents of the man we are punishing is important for
everyone, because a monarch who is persistently cruel or indifferent to the consequences their policies will have for other people generally turns out to be bad for
everyone, including the nobility and the smallfolk alike. Maegor the
Cruel, Rhaenyra's great-great-uncle, was so named for a reason, and he's still just barely within living memory (there being at least a few octogenarians and nonagenarians kicking around in Westeros).
From the political argument, these two needs both co-exist and are balanced. While a few of the most 'conservative' of the nobles may grumble that a
proper monarch wouldn't worry about the poacher's family and how if the poacher wanted his family to be provided for he shouldn't have poached and if his daughter starved he should just beget another...
[spits in disgust after trying to get into the headspace of someone comparable to Randyll Tarly]
...Anyway, while
some of the nobles may grumble about this, nearly all nobles will on some level respect a decision that balances upholding their rights with trying in good faith to limit the impact of the judgment to the man who committed the 'offense.' The commoners, meanwhile, take it as a signal that we at least
care and aren't actively going to crush them into jelly without even noticing, which is important, because some kings and nobles genuinely do that.
In the North, where the social contract between nobles and commoners is a bit stronger and places higher obligations on the nobles to administer and maintain social welfare via shows of generosity (note that the Starks appear to have just such a system in place at Winterfell), provision for the survival of the poacher's dependents signals to
everyone that we are able and willing to uphold such a North-style social contract, which is probably (I hope) going to win us respect up here.