Are exchanges any use? Things like Coinbase seem very agressive in knowing personal information about me, asking for a scanned passport, and that's a big no-no for me.
Are there other alternatives to Coinbase where I can just buy a bit of BTC with my credit card, and preferably without the intrusiveness of Coinbase?
alternatives exists and i believe there is even one built into the android wallet app mycellium but most will require a lot of verification, your only other option really is to use local bitcoins or a similar service to buy from someone near you, but this usually comes at an increased cost.
i actually cant think of any way off the top of my head to buy coins without full registration or personally knowing someone with them already, there used to be ways to get them using uKash or paysafecard and even second life linden dollars but i think theyve closed off all those various holes now
Local Bitcoins should have been your first (and only) answer. Sure it is kind of cyberpunky "dangerous" and you need to talk to strangers, but that's what Bitcoin is, its a feature.
Without all the 'KYC BS', it would be too easy for someone to use stolen credit card numbers to buy Bitcoin which would quickly bankrupt the service or at least cause them to have to pay high processing fees or have their processing terminated.