Prometheus does support push. It's just that it's considered such an antipattern that it's been moved into a separate module (the Push Gateway) that you need to run separately.

Pulling has a few technical benefits, though. For one, only the puller needs to know what's being monitored; the thing being monitored can therefore be exceedingly simple, dumb and passive. Statsd is similarly simple in that it's just local UDP broadcast, of course, which leads to the next point:

Another benefit is that it allows better fine-grained control over when metrics gathering is done, and what. Since Prometheus best practices dictate that metrics should be computed at pull time, it means you can fine-tune collection intervals to specific metrics, and this can be done centrally. And since you only pull from what you have, it means there can't be a rogue agent somewhere that's spewing out data (i.e. what a sibling comment calls "authorative sources").

But to understand why pull is a better model, you have to understand Google's/Prometheus's "observer/reactor" mindset towards large-scale computing; it's just easier to scale up with this model. Consider an application that implements some kind of REST API. You want metrics for things like the total number of requests served, which you'll sample now and then. You add an endpoint /metrics running on port 9100. Then you tell Prometheus to scrape (pull from) http://example.com:9100/metrics. So far so good.

The beauty of the model arises when you involve a dynamic orchestration like Kubernetes. Now we're running the app on Kubernetes, which means the app can run on many nodes, across many clusters, at the same time; it will have a lot of different IPs (one IP per instance) that are completely dynamic. Instead of adding a rule to scrape a specific URL, you tell Prometheus to ask Kubernetes for all services and then use that information to figure out the endpoint. This dynamic discovery means that as you take apps up and down, Prometheus will automatically update its list of endpoints and scrape them. Equally importing, Prometheus goes to the source of the data at any given time. The services are already scaled up; there's no corresponding metrics collection to scale up, other than in the internal machinery of Prometheus' scraping system.

In other words, Prometheus is observing the cluster and reacting to changes in it to reconfigure it self. This isn't exactly new, but it's core to Google's/Prometheus's way of thinking about applications and services, which has subseqently coloured the whole Kubernetes culture. Instead of configuring the chess pieces, you let the board inspect the chess pieces and configure itself. You want the individual, lower-level apps to be as mundane as possible, let the behavioural signals flow upstream, and let the higher-level pieces make decisions.

This dovetails nicely with the observational data model you need for monitoring, anyway: First you collect the data, then you check the data, then you report anomalies within the data. For example, if you're measuring some number that can go critically high, you don't make the application issue a warning if it goes above a threshold; rather, you collect the data from the application as a raw number, then perform calculations (e.g. max over the last N mins, sum over the last N mins, total count, etc.) that you compare against the threshold.

In practice, implementing a metrics endpoint is exceedingly simple, and you get used to "just writing another exporter". I've written a lot of exporters, and while this initially struck me as heavyweight and clunky, my mindset is now that an HTTP listener is actually more lightweight than an "imperative" pusher script.