(For this sentence, it sounds like you need an adverb rather than an adjective.)

Adjectives with a negative meaning:
stubborn
obstinate
recalcitrant
contrary
contumacious

Adjectives with a positive meaning:
unyielding
staunch
persistent

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Note added at 1 hr (2023-10-14 01:53:08 GMT)
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"cantankerous" is used to refer to a person (typically elderly) who is in a bad mood or is argumentative, and does not indicate that the person is doing or refusing to do some action.

It sounds like your two sentences could be translated something like the following:

1. "Mr. Zhang obstinately keeps playing video games." (negative connotation) or "Mr. Zhang obstinately refuses to stop playing video games." (negative connotation, and if someone or something is actually encouraging him to stop)

2. "Mr. Zhang plays video games unremittingly/unceasingly." (neutral connotation) or "Mr. Zhang just keeps playing video games without end." (negative connotation)

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Note added at 6 hrs (2023-10-14 06:45:17 GMT)
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Yes, "unceasingly" can be either positive or negative: "He bothers me unceasingly" vs. "He sacrifices himself for the sick unceasingly".

If you want to clearly praise him for not giving up on playing video games, then you could also say that he plays them "indefatigably" (a positive way of saying "untiringly" or that he "persists tirelessly").

If continuing to play video games is difficult for him but he successfully overcomes that difficulty, you could also say that he "continues resolutely" or "carries on steadfastly" or that he plays them with "unswerving devotion".