The transaction from "Tempered bitcoin Core" would be signed right, because it would have all private keys of the user that runs the app. User would think that because he got BitcoinCore from official site and checked hashes it is OK, .. BUT it will not be OK!
All people that defend open source here claim that "The open source code given compiles to version given". That assumption could not be true. If (in worst case scenario) main ones or all proggrammers decide to sabotage they can get the given oipen source, than add the code for any exploit than give the .exe to public and cash the transactions for one day or one hour in Fiat and all that would be irreversable and ruin trust in btc.
I understand that most people here want to trust, but now the trust is blind trust. Trust that anyone else has checked all and all is fine.
In my oppinion giving so much power in the hands of so many few (programmers) is really unacceptable. There must be other way to insure reliability.
We do not want to trust a country or a government, but we trust 5-10 programmers. They are fine so far, ... but !!! We are working with billions dollars worth now! And people are corruptable! The temptation of billions $ is too strong for average person or group of people.
About >50% computing power. As I know there were times when biggest pools did more than 5 consequtive blocks - 6 blocks are enough to add transfers of money that did not initiate transactions (not talking about double spending, but about unauthourised transactions in system).
No, you can't simply
spend coins unauthorized.
That's not possible.
Without providing the signature or more generally, the unlocking script together with its parameters, an output can not be spent.
I mean.. anyone could create a block which spends such an output without being authorized.. but this block would be invalid and therefore not accepted by every other node. Regardless of whether they have more than 50% hash power or not.
They would fork away from bitcoin and no one would care.