This! I've not had any issues with Life360. I only have two locations, home and work, my kids and wife have it on their phones. ST has always done the things I've asked of it when the phones leave or come back. All android, SG5, SG3, MotoX 1st Gen and MotoG's first gen.
You should never use any battery saver apps or set the built in battery saver if you expect behind the scenes GPS tracking to work right. GPS is a low drain device if implemented correctly compared to wifi and cell radio power drain so it shouldn't have a big effect for Life360.
For me, the ST presence has sucked from day 1, it basically never worked, maybe 2 times did it actually notice when I came home. I don't believe the ST app is using the GPS correctly AT ALL.