Reading the original Satoshi paper, the point of Bitcoin is to be decentralized/trustless and to enable microtransactions by lowering transaction fees. Both these goals seem to have failed in practice, the latter spectacularly (although other cryptocurrencies have done better in this regard). To truly be trustless, you can't generate keys/addresses on some website like bitaddress or use some other online wallet, as y…
Bitcoin rose out of the previous financial collapse by appealing to people who have lost trust in institutions. The problem with building a trustless answer to that problem is that (assertions ahead) life isn't compatible with an absence of trust.
Trying to build a trustless financial system is a quixotic undertaking at best. It really smacks of a utopian dream where fundamental aspects of human nature are blithely ignored.
Of course, as you point out, no one is really trying to do it. Most people will have to trust Coinbase or whichever exchange they choose.