I had the same issue. Used the on-line chat feature to talk to Obi help desk. For me, the issue was two-step authentication. Once I turned two-step authentication off on my Google account, my phone started to get the proper stutter and visual blinking light indicator.
Interestingly, the help desk did say that the only times an Obi talk device will indicate you have a vociemail is
a) rebooting the device
b) making a phone call from your connected phone
c) automatically once every hour
So if someone leaves a VM and "a" and "b" have not yet happened, then you won't get an indication for up to 60 minutes.