IOS – Is it possible to make Apple Watch Series 5 measure heart rate (pulse), passively, all the time

If the watches are different, and the phones are different, then the only thing that appears to remain the same is you.

You don't mention, but have you ruled out the possibility of there being anything about you? For instance, I remember reading that tattoos could prevent the heart monitor obtaining a clear reading, and there was some talk about people with very black skin also having difficulties. Example article here http://www.bbc.co.uk/news/technology-32537030

The other possibility is whether there is something with the tightness with which you wear it?

I imagine the WATCH will keep as much data as its onboard storage will allow, which is several gigabytes. So, caching Health data really shouldn’t be an issue for most intents and purposes.

Since the title of the question is about the disadvantages of Airplane mode, know that placing your WATCH in airplane mode will preclude you from connecting to the Internet. Therefore, any feature that requires an active Internet connection will not work. Siri comes to mind.

Finally, since the Bluetooth antenna is also disabled, you will be unable to connect a set of Bluetooth headphones, or an external heart rate monitor. Any app that requires an iPhone connection will not work either, meaning all watchOS 1 apps, as well as many, if not most, watchOS 2 apps. GPS via your iPhone will also cease to function.

Enabling Airplane mode thus disables most of the features of your WATCH. On the other hand, there aren’t too many advantages to enabling it. You might see some gains in battery life, though most of that will come from not being able to use your WATCH for a whole lot of things. But if you weren’t using your WATCH a lot to begin with (outside of activity tracking), then battery life is more than adequate to last you an entire day.

Even if you are on a marathon hiking trip, you will defeat much of the purpose of tracking that workout anyway, since the lack of GPS will result in less accurate distance measurements. In that situation, it would be preferable to turn on Power Saving Mode under Apple Watch → Workout.