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The problem is whoever is making the decision about injecting more or less money into the system is playing the exact role the Federal Reserve is doing right now. That group or individual has the ability to manipulate the system and now you are back to the same problem fiat has.
Not if, as I said, the mechanism to inject Money into the system was built into the coin. In a completely mathematical way. No humans would be used.. utterly beyond repute and corrupt manipulation. ( We can worry about THAT system at a later date..
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Also I don't agree you need to add money to the system as the economy grows, Bitcoin's beauty is in its ability to be divided down to 8 decimal places. As a Bitcoin becomes more valuable you have to use smaller and smaller amounts for daily activity, all without the risk of inserting someone into the system that can manipulate it.
This still does not address the inherent deflationary nature of Bitcoin. As anyone who is currently hoarding their coins knows, there's very little that could tempt you to spend them, as they go up in price, and basically, always will. This has it's own set of economic problems.
In my, admittedly short, economics education, as I am mainly a mathematician and programmer, it has come to my attention that deflationary currencies have just as many problems as inflationary ones ?
Bitcoins are, if they survive, an excellent Store of Wealth. A currency for day-to-day use, as a medium of exchange, seems to require quite different properties and more importantly service a different niche in the market. I'm not sure it is possible to do BOTH things well.
But even if it mathematical, it still has to be programmed in by humans, intervention via an algorithm is still intervention.
You cannot hoard bitcoins, or any currency indefinitely, you would have to spend on day to day items, and investments that came along would get you to spend them as well.
Inflationary currencies punish saving, and it is only through saving and investment that new innovations and products come about. You are correct both kinds have problems, but intervention is not the solution otherwise central planning style systems should work, but in practice they don't.