Solution 1:

I love the word gamify. Before that we used to have to use the set phrase, "Turn it into a game". My mom would say, "Okay kids. Let's turn it into a game." Then she would gamify our chores with little tiles for points and score cards for the week, with minimum number of tiles to earn for a popsicle, etc.

Your example sentence:

Our parents and teachers used to turn our routines into a game , rewarding every useful contribution and penalizing unproductive actions. Today, it's known as gamification!

Solution 2:

In the UK, the term "Brownie points" had pretty much the same meaning since the 1960s and was quite widely used. I always assumed the origin was from the "points and badges" awards in the junior division (Brownies) of the Girl Guide movement.

Wikipedia has some alternative etymological conjectures: https://en.wikipedia.org/wiki/Brownie_points

(But it doesn't quite fit the OP's "usage sentence", since the noun "Brownie points" never got verbed.)

Solution 3:

Jean-Francois Lyotard wrote about the "sportification" (presumably the verb he used would have been sportifier) of society, in which things are stripped of their traditional character in the name of points, competition, and winners. Consider the way that "sports" themselves evolved from activities with a distinctly moral character (compare German academic fencing with sport fencing, for example, or English public-school ball games before and after the creation of the Football Association). It's a slightly different phenomenon though, and at least in Lyotard's eyes, strongly pejorative.

The Lyotard book in question is The Postmodern Condition, French edition 1979/English translation 1984, but it's been many years since college and I can no longer say how much of the above reflects Lyotard and how much reflects secondary literature on him (not to mention my own thoughts on the matter). But the word does appear there.