Our prayers have been answered as indicated by this MDSN blog post:
SQL Server 2019 introduces a new message, with additional context
The error should be much more meaningful now:
Msg 2628, Level 16, State 1, Line 14 String or binary data would be
truncated in table 'DbName.SchemaName.TableName', column 'Col'.
Truncated value: '...'.
Also, the good news is that this change will be back-ported in both 2017 and 2016:
This new message is also backported to SQL Server 2017 CU12 (and in an
upcoming SQL Server 2016 SP2 CU), but not by default. You need to
enable trace flag 460 to replace message ID 8152 with 2628, either at
the session or server level.