[Notif redesign] Use app icon cache in NotificationInfo

It's not ideal that we need to call this from view code, but it's still
an improvement over the status quo since there's a high chance the icon
is in the cache already and we don't actually need to load it (unless
we're showing the small icon for the notification).

Bug: 394822197
Test: tested manually
Flag: com.android.systemui.notifications_redesign_guts
Change-Id: I87c3e5f1a90248c83adf4f71486e9cf8156d9522