How can I make it as easy as possible for someone to make an emergency call?
Apple Watch Series 4 provides a 1st party solution for the aforementioned requirements.

You can consider an Apple Watch Series 4 or later with Cellular connectivity. Apple Watch Series 4 and later feature build-in fall detection mode that can also alert the emergency services and customizable emergency contacts. A cellular variant remains connected to the Internet even if the paired iPhone is not in the proximity.

You can refer to the Apple Support document, Use fall detection with Apple Watch to learn more:


  If Apple Watch Series 4 or later detects a hard fall, it can help connect you to emergency services if needed.
  
  If Apple Watch Series 4 or later detects a hard fall while you're wearing your watch, it taps you on the wrist, sounds an alarm, and displays an alert. You can choose to contact emergency services or dismiss the alert by pressing the Digital Crown, tapping Close in the upper-left corner, or tapping "I'm OK."


and


  If your Apple Watch detects that you're immobile for about a minute, it begins a 30-second countdown, while tapping you on the wrist and sounding an alert. The alert gets louder, so that you or someone nearby can hear it. If you don't want to call emergency services, tap Cancel. When the countdown ends, your Apple Watch automatically contacts emergency services.


The fall detection and emergency calling feature is built into Apple Watch and doesn't require you to install a separate app or get a separate subscription.

Even without a fall, Apple Watch makes it very easy to contact the emergency contacts which can be configured. Having a Cellular model gives the extra peace of mind that the Watch would have cellular connectivity even if the paired iPhone is not in range.

Since the features comes built-in from Apple, there's no data privacy concerns.

Apple Watch Series 4 and later also supports "Hey Siri".