It's just a terminology issue. The word "random" is vague and its use should in general be avoided. For example, for a lot of people "a random situation" means "all events are equally likely" -these people would have a hard time understanding why "random arrivals" follow the Poisson and not the Uniform. Sometimes the word "random" is used as a "verbal substitute" for the word "independence".
In most cases, the Poisson distribution models essentially the allocation of probabilities over the possible values of a sum of independent indicator functions, (and this is why it is so closely related to the binomial distribution). To obtain the Poisson, we need to make two assumptions related to stochastic independence:
a) That each indicator function (will I call the call center or not in the current minute?) is independent of all the others (whether you call does not affect the probability that I will) (So for example, when the service is down, you can see that calls to call center will stop being independent from each other, since calls will have a common source)
b) Each indicator function, viewed as a stochastic process, is independent of its own past: whether or not I called the call center one minute ago, does not affect the probability that I will call in the current minute. (for this to be realistic one should decide on the length of the interval juidiciously -which is part of the applied art of working with Poisson: it is stupid to say that if I called the last minute does not affect the probability that I will call in the current minute, but perhaps, if you set your time period as "day" or "week" the Poisson becomes acceptable).
It is these two assumptions related to independence, that people express by the use of word "random" in this case, which is very confusing indeed.
Now, what would it mean to say that "arrivals follow the uniform"? It would mean that it is equally probable that we will observe, say, 1 arrival or 1.000 arrivals. Or that it is equally probable that out of the $N$ indicator functions, only one will take the value unity, or that 1.000 of them will take the value unity. Or any other value from 1 to $N$, for that matter.
So what kind of random variables are these, that their sum follows the uniform?