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Exodia, for those unfamiliar, lets you win the game when you have all 5 pieces of it in your hand

Maybe from dirt or smudges while transferring it from your bag to the receiver

Yes, on top of being reasonable extenders, these monsters each also boast an interactive effect, kind of like Therion (Coincidentally, another engine that might fit here)

Keep in mind that removing a card from the chain won't necessarily negate it's effects

Eternal Sunshine X TRAP Continuous Monsters you control gain 500 DEF for each "Ancient Fairy Dragon", and monsters that mention it, that you control

This card can discard itself to send a Cyber Dragon from the deck to the graveyard to Special Summon itself and get you a Spell or Trap search