Forward them to whom?
If there's no way the phone knows in advance that any given message is spam or phishing, then how is it supposed to know where to send it to?
Additionally, even if there were some mechanism to detect this, the target company (e.g. your bank, Apple, etc.) has no way of responding to it - what do you expect your bank to do, for example, when they're notified that you received a phishing text? They can't prevent the messages from going out because they're not the ones sending it.
Even if one company 'solves' this problem, it would need to be universally applied in order to be effective - it's not much good if BankA build a response system if BankB and BankC don't have something similar - it will just generate a false sense of security because you assume that your bank is secured just because some other bank is.
Ultimately the whole spam/phishing issue is an industry-wide problem that requires an industry-wide solution. Ultimately I think this has to be carrier-level since they are the last mile (i.e. the ones that actually deliver the message), and so have the best visibility and chance to fix the problem. It does introduce all kinds of questions of privacy (for example, are you OK with the carrier looking at every text message you see just in case it's a phishing text?)