Default app state inaccurate while syncing

### Steps to reproduce

Sync a large account. Observe the client shows misleading content to the user as if their rooms/account don't exist at all:

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Reported by @InezMc

### Outcome

We shouldn't display content which misleads the user into thinking all their data may be missing. We should indicate loading consistently as some tabs show a spinner and others don't.

### Your phone model

_No response_

### Operating system version

_No response_

### Application version

_No response_

### Homeserver

_No response_

### Will you send logs?

No

Start by reproducing a large-account sync in the iOS client and compare the tabs that show a spinner with those that display misleading empty content. Trace the default app state and loading indicators used by those tabs. Done means syncing consistently presents loading feedback without implying that rooms or account data are missing.

Written by the indexing model from the issue text.