because the node has to fully verify any transaction that it accepts
Why? If you are sending some BTC from one address to another and it is correct, then there is no need to check if ALT inputs and outputs are correct. If both parties will sign this transaction, then it means that both parties checked that everything they signed is correct.
checking the digital signature is only one part of the transaction verification. the other part is to check if the "coins" being spent even exist or not.
think of it as when you write a check. visually verifying if you signed the check with your real signature is only one part of the verification, the other part is the bank checking if you have the balance you are trying to give someone else.
that second step in bitcoin (checking balance) is done by checking if the inputs of a transaction exist and weren't spent before. so each full node has to have downloaded and verified the entire blockchain to create a database of all unspent transaction outputs.
And on BTC blockchain no miner have to worry about some ALT chain correctness, because it will be checked on this ALT chain by their miners.
the idea of decentralized bitcoin and similarly any decentralized altcoin is that you don't trust or rely on anybody else including miners or even other nodes. you download and verify everything yourself.
and when you can't verify a tx fully, it becomes possible to create fake transactions
Yes, it will be possible to create fake ALT inputs and outputs, but in such case it will not be signed by another party! I mean, miners will not check ALT inputs correctness, but they will check that both signatures are correct.
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miners never check anything including signatures. nodes do.
as i said above the point is that a node has to be able to verify everything otherwise things become centrailzed. and you can never trust what other node tells you. it might be malicious and give you fake blocks and transactions from the other chain.
