With a custom crash reporting system (like the ones specialized at Ask the user to send crash log after crash on iPhone) to send the log, the app needs to restart. Why? Isn't there a possibility to send it during the custom exception handling? Or is there a crash reporting system that doesn't need to restart the app?
When a crash occurs the app is in a highly unstable state. So a crash reporting library can not do anything since even allocating memory at crash time may cause way more damage. So crash reporting SDKs can only use so-call async-safe C methods to collect all crash data. Any Objective-C code can not be processed either and also the iOS networking stack can not be used.
Please also note that exceptions are just one example of app crashes in Objective-C, there are also crashes triggered by low-level BSD signals. Both types mean that the app is in a highly unsafe and unstable state and as little code as possible should be invoked at crash time.
So this would require a rewrite of most of the networking frameworks to be able to send data out at crash time, and this may even not be possible to do in a safe way. That is why all proper crash reporting SDKs don't do anything like that.
In addition, on iOS it is not possible to create another process which could send the data in the background, so the only safe and possible solution is to send data the next time the app starts.
Now this has another conclusion that crashes that happen early on app startup might never be sent since the app is crashing before or while it sends. Some SDKs provide mechanisms to handle that scenario which most likely requires changes in your app startup code.