I’m working on autometrics (https://github.com/autometrics-dev/autometrics-rs) and people asked whether it could be used in embedded contexts but I wasn’t sure how you’d hook up the device to something like Prometheus.
Most would probably end up as pushing metrics via... something (MQTT?) to gateway that will push it to their metrics system.
Pulling/polling metrics honestly is pretty bad idea most of the time. It makes it easy to implement ("just* provide a web page with metrics) and allows poller to control intervals (which is only a problem if your configuration management isn't up to snuff), but complicates a lot, now you need to setup whole endpoint discovery instead of just saying "hey, apps, here is URL to push your metrics to, have fun".
We for long (10+ year) have system based on collectd and it's line protocol, and while it have some drawbacks (no tags, just rigid structure with host->plugin->plugin_instance->type_type instance, works well most of the time but 100% of the time), push nature is very light on resources (it's just an UDP packet full of metrics, no need to setup whole TCP connection, doesn't even need bidirectional firewall)
In my experience the discovery and configuration is needed anyway to do something with the metrics.
ARM chips intended for embedded uses also have a single wire output peripheral for tracing and similar purposes that could be used for this.