This is a normal use of "would" to convey Ellen's intention indirectly and humorously. "I think we will play a game before you get too drunk" would be too direct and realistic.
By using "thought" and "would", Ellen is making it clear that she is joking rather than making a serious statement about people getting drunk. "Would" and "thought" imply that Ellen's "thought" is hypothetical rather than real.
would = could possibly -> it is the present conditional ->
"So I thought we would play a game, if you want to play a game, before you get too drunk".
"So I thought we could possibly play a game, if you want to play a game, before you get too drunk".
"So I thought we might play a game, if you want to play a game, before you get too drunk".
Would is used in these ways:
1 -> I wish he would do what I asked. -> Regret about somebody's behaviour
2 -> Would you follow me, please? -> Polite request
3 -> He said he would do it tomorrow. -> Reported speech (past tense of will)
4 -> I would rather you didn't do that. -> Expressing a preference about an action
5 -> And every day he would go to work by bicycle. -> Past habit
6 -> Well, he would say that, wouldn't he? -> Commenting on somebody's typical behaviour.
7 -> Would you tell me the answer? -> Indirect question
8 -> Would you like to go to the cinema? -> Polite offer
9 -> I would imagine we'll arrive at about six . -> Softening an opinion you are not sure about
10 -> He wouldn't do what I asked. -> Refusal in the past
11 -> One day he would return to this university as a professor. -> Future in the past
12 -> If he had known this he would never have asked her. -> Third conditional
13 -> Be careful! It would fall over if you touched it. -> Predicting a high probability.
14 -> etc.
Will is used to indicate
(i) a present intention
I will buy a car -> present intention
(ii) with if clauses, the future (predicted) action or state, of someone or something.
If you build the wall too high, it will fall down - future action
If stop building the wall now, it will be too low. - future state of something.
If he continues eating he will become fat. - future state of someone.
Look at the way she is eating those biscuits! That is her fifth -> she is eating it now.
Look at the way she is eating those biscuits! That would be her fifth. -> if she starts eating it
Look at the way she is eating those biscuits! That will be her fifth. -> when she starts eating it
Look at the way she is eating those biscuits! This would have been her fifth. -> if she had eaten it.
The sentence itself is in the past tense, thought being the main verb. But the modal verb would in “would play” implies a condition rather than indicating the past tense. It’s used to make a suggestion: I thought we should/could/would/might play a game [if you want to]?