Randori practice, as such, is very possible in this art. I spent a fair amount of time in a place that uses randori almost exclusively as it's base of practice and did so immediately. The idea that you can't do this in Aikido because it's TOO dangerous is wrong.
How did you work with a beginner for instance? Simple, you didn't throw them in dangerous ways and went slow with them. Basically beginners were full-on blending practice in a moderately controlled way. As they became more advanced the practice evolved into whatever it evolved into. More than a few times, with the right partners I'd wind up wrestling on the mat. I've even tackled the sensei when called up to uke.
Paul is exactly right.
A couple of notes. I personally don't subscribe to the above approach. I think there needs to be a structural base, particularly for raw recruits. Without one they pick up as many bad habits as good habits. I don't, however, subscribe to the idea that it takes years to achieve. If it takes years to get there (and I'm thinking a decade or so as some do it) my belief is that moving to this flowing practice will be very difficult at best. I honestly think it's almost impossible at that point but recognize there are exceptions.
Second, the dojo mentioned above was definitely taking a more spiritual (whatever that means) approach. This was actually a source of frustration for me as there were really only a couple of people interested in playing more physically. Still, randori is very doable and I've done plenty of it. For what it's worth, I've been bashed both in structured and unstructured environments too.