It's your game. If you decide players can have multiple inspirations, but only use one at a time then that's the rule. RAW is only a suggestion.
... yes ? I suggest to use it too, giving you my idea
I'd argue you could even do away with the action if you like.
Probably yeah.
after all, the seed of inspiration often takes a long while to germinate, and only under the right circumstances.
I think it's a good idea, not dissimilar to WBW's Potion of Advantage. Ultimately it's up to you, as it is every DM, to decide if you want players to be able to stack inspiration in that manner, but as you pointed out, allowing players to stock inspiration tends to discourage them from using it unless they have a stockpile already, because they don't lose anything from not using it.
What I did was use tokens for inspiration and saying "you can have as many as you want and then cash in all of them to get advantage". That worked well.
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