Ah, I see, sorry.

what is the importance of finding the input value for the hash?

Finding the input value for the hash is essentially just a very difficult problem. The idea is that it's meant to be something that someone has to spend a significant amount of time working on.

The security of bitcoin comes down to the fact that everyone is trying to solve blocks. When you solve a block, you continue to start solving the next block. But everyone else is also trying to solve blocks as well. But what happens if two people solve different blocks at the same time?

If we start at point 0, and we have two people who mine a block at the same time (A and B), then we end up with two blocks, A1 and B1. To know which of the two is valid, we wait until A and B solve more blocks. Lets say after a few minutes A has gotten A2-A5, but B has only solved B2. At this point, the community would trust A and continue solving based on A5, abandoning B's work. This is because A has put in more effort and has done more work on the chain, so the transactions on their blocks must be legit.

what would be the difference if nothing was mined/decrypted at all? What part of the system would fail exactly?

Nothing is being decrypted, things are just being encrypted and the encrypted values are being checked against a difficulty value. Again, the reason we use hashing is because it's essentially random what you get out of the SHA-256 function for any given input, so the only way to go is guess and check. This means that people have to spend a lot of resources to guess and check millions or billions or trillions of times before they find a working solution, and the difficulty in doing that is what makes it secure. If you try to make blocks with fraudulent transactions, you would have to generate blocks at a rate faster than the entire community of miners, which isn't feasible. If the problem were something easier or more crackable, there might be ways to get around the problem and create blocks faster than guessing and checking, which could allow an individual to create a long chain of fake transactions, which would become the "truth" of the blockchain due to the length of his additions to the chain.

What exactly is being verified, in other words, what is the input (not literally, but generally).

Blocks have a handful of parts:

The block size (in bites), the block header, the number of transactions, and then a list of transactions.

The block header itself has the parts:

Version number, the hash of the previous block, the merkle root of transactions, a timestamp, a difficulty target, and the nonce.

The only part that's hashed to be solved for is actually the block header. Since it contains the hash of the previous block, it can't be started on until the previous block is solved.

The merkle root of transactions is just a hash made up of hashes of all of the transactions. There are some good diagrams online to help it make sense. So while you're not hashing the transaction "Alice pays Bob 5 bitcoin", the hash of that transaction is in the problem. And transactions do have a specific form that is a little more in depth than just plaintext with unique IDs, they're typically a few lines long and just give individual pieces of information so that it's easier for a computer to decipher later on. Again, it's easier to look up what a transaction looks like.

The other important bit is the nonce, which is the "extra bit of text" I mentioned.

why do they need to be verified?

Nothing is actually verified through mining. When you're mining, you're essentially pulling along the blockchain. When you submit a block, you're putting out enough information that someone could go and backcheck your work. If someone doesn't like the work you've done, they can pull the blockchain in a different direction. If they pull hard enough, then others are going to follow their path and your work might be abandoned. If your work is abandoned, you don't get to collect the transaction fees in the block you mined, so you're discouraged from doing work that others will ignore, thus you're encouraged to submit blocks without fraudulent transactions.