There is a hadith regarding remembering Allah, where Imam Ali as states that “anything which preoccupies you from remembering Allah is considered gambling”. I may have narrated incorrectly but, I do know that the gist is that being in a state where you don’t remember Allah is considered gambling.

Is this hadith sahih and would it make playing video games or watching movies become haram?

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The gist of this hadith is that anything that distracts you from:

1- Allah
2- The obedience of Allah
3- Good deeds

is to avoider because it becomes a burden and it will be a source of regret for us on the Day of Judgment. So any activity you do, make sure it’s purposeful and beneficial. Sometimes we need leisure time to be refreshed (like take a nature walk), that is ok. But to be addicted to modern games is very destructive. Now if one plays a halal game occasionally (to be refreshed if one is tired for example) it’s ok, but there are better alternatives than just playing such games.