But I googled and it seems with some extra verification steps they still accept users from Germany. I'll contact them and make sure.
GDAX is 6 years old and seems extremely legit. It'd be exactly what I'm looking for.
If not GDAX then I might check out Bitstamp. They're licensed by the Luxembourg government and they've been around since 2011 (basically the last early exchange left standing). I have trading friends who have been using them for years without any major issue. They do SEPA transfers. They used to convert all EUR deposits to USD to force all liquidity onto the BTC/USD market. I see they now have a BTC/EUR market, but I'm not sure how liquid it is.
One note on Bitstamp, though. Their AML/KYC procedures seem to differ from other exchanges. Some people report that they have been subjected to KYC questionnaires about sources of funds, etc. And some people have also had their accounts terminated without explanation. I think this is a pretty small minority, but just FYI.
Regarding Kraken....I've had bad experiences with site and API downtime there. I'm also less comfortable storing funds there. They don't hold any licenses and they deal with Tether.