The two apps aren’t comparable in that way, because they’re doing different kinds of location tracking. Using the approach Google Timeline does, it would be impossible to achieve anything like Arc’s level of detail and accuracy.
Google Timeline uses what’s called “significant location updates”. Using significant location updates means that the app is suspended completely between location updates, and only receives new location data once the device has moved maybe 100 to 150 metres or more. The app is then woken up, the location data delivered, and then goes back to sleep. That allows for very low energy consumption, but makes detailed path recording impossible. It’s incredibly low detail.
Arc instead receives a constant stream of location updates (except when stationary and in “sleep mode”). That constant stream is the only way to achieve detailed recordings, but it means that the app is never suspended in the background.
Arc also can’t allow itself to be suspended even when in “sleep mode”. Arc needs to be able to start actively recording again almost immediately, once you leave the place you’re at. If the significant location updates service were used, the first 100-150 metres or so of each trip would be lost.
That would also make it impossible to detect short trips between nearby places. For example if you pop out to go to a nearby convenience store, only 100 metres away, Arc can accurately record the walk to the store, the visit to the store, and the walk back home. Google Timeline can’t record those short trips, and is also likely to completely miss the fact that you ever left home!
Basically there’s a huge rift between apps like Arc and apps like Google Timeline. If Google Timeline wanted to record more detail, even if not as much detail as Arc, it would need give up on its strategy of using the significant location updates service, and that would mean using basically the same amount of energy/battery as Arc. There’s not really any middle ground between - it’s one or the other.