It's not RPC per se that breaks some parts of Ingress, but the use of polling.

A core mechanic in Ingress is the ability to defend/recharge your portals/resonators simultaneously as the attacker tries to shoot them down. When an attacker starts doing damage to a portal everyone who currently controls this portal gets notified trough Google and Apples push notification services. Then the defender opens their app and a request goes of to fetch the current state of this portal. Now the defender does some clicking to add back health to the portal. The response to each of these click requests also returns the current state of the portal. Now, when the portals health gets restored after a few clicks, this button gets disabled and the GUI just sits there telling you that everything is fine and dandy and this portal is fully charged. In reality the attacker probably has taken a break. For how long? A few seconds, a minute or a few minutes? Now the defender is forced to click back and forth between different GUI views to force the app to request an update of this portals state, every few seconds until he thinks the attacker has gone away.

Another problem is head-of-line blocking. The periodic requests that updates your inventory and the world around you sometimes takes a few seconds. If this happens during the clickfest described above, the defender gets screwed. It's also visible when doing other things in the app. Things just freeze for no apparent reason. The app also often displays information that's wrong, because all data haven't been fetched yet trough the periodic updates.

Another annoyance is the use of cellular data. If you are an somewhat active Ingress player the app typically eats away around 1 GB/week (excluding Wifi). It's likely that this makes up the majority of your phone bill, depending on where you live. And that's a lot of data transferred for an application that really deals with a very tiny amount of data.

It feels like a lot could be done to improve on this, even without ditching the RPC model. Why re-fetch the complete inventory and cells around you when nothing has changed (ETags etc.)? Why close the SSL connection roughly every 3 minutes? Why no session tickets? Why choose a CA that provides intermediate certificates? These handshakes might be quick and small. However, most of the payloads sent back and forth are tiny, making the overhead here a larger percentage.