I had a miss click while playing arena and played a search card and it automatically showed me my deck. In the paper magic could you play a search card and choose not to look at your deck if you had scry something to the top and you just want another affect of the card, I had Aetherflux Reservoir in play and mainly just wanted the life gain from playing a cheap spell.
Yes. (Unless the card says "You may search your library for ... If you do, shuffle it afterwards.)
608.2c The controller of the spell or ability follows its instructions in the order written.
If the spell tells you to search your library, you search your library. Most of these place restrictions on what cards you search for; in those cases, actually finding a card is not required:
701.19a To search for a card in a zone, look at all cards in that zone (even if it’s a hidden zone) and find a card that matches the given description.
701.19b If a player is searching a hidden zone for cards with a stated quality, such as a card with a certain card type or color, that player isn’t required to find some or all of those cards even if they’re present in that zone.
701.19d If a player is searching a hidden zone simply for a quantity of cards, such as “a card” or “three cards,” that player must find that many cards (or as many as possible, if the zone doesn’t contain enough cards).
In the specific instance (from OP's comment that the card was Enlightened Tutor):
Search your library for an artifact or enchantment card and reveal that card. Shuffle your library, then put the card on top of it.
All these are mandatory actions. Glorfindel gives the rule that says you may "fail to find" (701.19b applies; "artifact or enchantment"). If you do fail to find,
609.3. If an effect attempts to do something impossible, it does only as much as possible.
Means that you won't reveal "no card", and after shuffling, you won't put "no card" on top, but it is possible to shuffle, so you do.
So here you may not choose to not search. You also may not choose to not shuffle (for obvious reasons!) This would also apply in paper, but it's likely "at the kitchen table" that everyone equated "fail to find" with "don't search", and didn't realize the effect that had on known cards in the deck (or shortcutted it frequently when it didn't matter, and didn't realize it's different when it does matter. I have certainly seen on YouTube that shortcut taken in Commander games - when everyone agreed that no card's places were known).
You are allowed to fail to find if you are searching a hidden zone. Your deck specifically is a hidden zone.
701.18b If a player is searching a hidden zone for cards with a stated quality, such as a card with a certain card type or color, that player isn’t required to find some or all of those cards even if they’re present in that zone.
You are required to shuffle your deck afterwards regardless if you failed to find or not.