It seems to me that only a ‘the’ could make almost 180 degree difference in meaning between:
Play games and play the game
Could I suggest that ‘the game’ refers to a definite game with certain rules acknowledged by all players so as to make ‘play the game’ mean ‘to act fairly and honestly’ whereas ‘play games’ implies a sense of reckless game playing (without ‘the’) so as to ‘be evasive or deceptive’?
There is a mixture here of reality and metaphor. Play games can simply mean what children do or what anyone does for sport. It can alsoi be used as a metaphor in the sense of not being serious about what you say or do - often used in the continuous form as in: You’re just playing games with me. I want a straight answer. A colloquial expression similar in meaning is ‘mess about’
The poet Henry Newbolt (1862-1938) wrote:
and here the comparison is made between playing cricket and fighting a war (the First world war) a very British attitude in those days. So the expression ‘play the game’ suggests today in an oldfashioned way the exhortation to follow the rules, behave correctly.