Life debts are a real thing in Wookiee culture and are indeed Very Serious Business. But not even the most strict of conservative would argue that a life debt requires you to drop everything and dedicate the rest of your life to the person you owe the debt too.
Kashyyyk is a dangerous planet. People save each other’s life all the time. What it does create is a lifelong bond and a sense of obligation. Strengthening of inter-group ties and cooperation, yada, yada…
So when Han saved Chewie from slavers, Chewie could have done the equivalent of exchanging business cards, saying hey I owe you one and maybe we could stay in contact, exchange the odd letter.
And gone on his way with the basic requirements of the life debt satisfied.
But the kid had obviously just lost everything and Chewie thought he should probably watch over him for a little while, just until he got back on his feet.
Which quickly turned into “Child, you need a keeper"
(as an aside, Chewbacca, no matter how much he like to project himself as the lone voice of reason, is terrible at keeping Han out of trouble, he agrees to stunts even when he should know better and has his own temper beside.)
and then into "He’s mine. You hurt him at your own peril."
10 years down the line there are a number of Wookiees, his wife among them, who are like:
"The life debt excuse stopped being believable 6 months in. Just admit you’ve adopted him and prefer galactic smuggling to life on Kashyyyk and stop giving outsiders distorted ideas about our cultural norms!”