You are certainly right that it becomes an M/G/1 model and thus that M/M/1 and M/D/1 will give reasonable approximations. The reason that it's an M/?/1 model is partly what you're saying, but also partly because the random assignment of the incoming requests acts as a random filter on a Poisson process. Poisson processes are just defined by not having history, and the random filter is not history-dependent either, so the output from the filter -- the input to any node -- is still a Poisson process.

What's interesting to me here is: Suppose rather than doing this with a random variable, you do it with a summing filter, a simple counter:

    on request r: 
        node[n].handle(r)
        n = (n + 1) % node.length

That's a really simple solution but it should tend to average out the Poisson input, which gives you something closer to a D/G/1 problem.