Why do you need a full node to deploy a smart contract? Can't you just send a transaction to the network from a wallet?

I just realized I kind of forgot how this works. When exactly and why do we need something like infura? I guess for reading the blockchain you need to talk to someone that has a full copy of the ledger in case you do not. But then what are the ideas there again about trusting that information? Can you trust it because you can get it from many different nodes and can just compare? Or what's the role of light clients again?

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Thanks in advance