What's the rationale of using ECDSA instead of RSA? One major drawback of (EC)DSA is that it requires a good RNG, otherwise the private key may be accidentally exposed. This is also one of the major reasons (I found so far) why no(?) smartcard supports any kind of DSA. In contrast to that, RSA, while usually being used for encryption, can also be used for digital signatures and behaves deterministic, i.e. signing the same message again will yield the same signature everytime. Lacking the requirement of a RNG (other than in key generation) this is why RSA is available on many smartcards, which would make bitcoin much safer without requiring a rather complicated offline-wallet setup.

So, does ECDSA provide anything bitcoin absolutely needs that RSA cannot achieve? Or is it merely bad luck in Satoshi's choice?
I'm not sure why you think RSA is much safer than ECDSA. As you can read here: https://crypto.stackexchange.com/questions/3216/signatures-rsa-compared-to-ecdsa 

ECDSA offers same levels of security as RSA, but with a much smaller footprint. 

In fact, the more you increase the security, the larger the RSA keys become compared to ECDSA.  This makes RSA less fit for a system such as bitcoin which requires small packets to be sent around the network all the time (being peer-to-peer).