Could an update on all statistics separately cause bad performance?
Question: I am not too familiar with stat maintenance, so I am wondering, is it really necessary to run a weekly job separately to update all statistics?

Yes, you need to maintain your statistics if you have large tables. Read here how Auto update statistics works: 


  Auto Update Statistics Option Statistics are checked before query
  compilation or before executing a cached query plan. Statistics are
  considered out-of-date when:
  
  There was a data change on an empty table. The number of rows in the
  table was 500 or less at the time of statistics creation and the
  column modification counter of the leading column of the statistics
  object has changed by more than 500 since then.
  
  The table had more than 500 rows when the statistics were gathered,
  and the column modification counter of the leading column of the
  statistics object has changed by more than 500 + 20% of the number of
  rows in the table when the statistics were gathered.
  
  A table in TempDB with less than 6 rows had at least 6 row
  modifications.


Question:  Don't statistics get updated after a certain amount of changes or during an index rebuild anyway?


  Rebuilding an index, for example by using ALTER INDEX … REBUILD will
  also update index statistics with the equivalent of using WITH
  FULLSCAN unless the table is partitioned, in which case the statistics
  are only sampled* (applies to SQL Server 2012 and later). Rebuilding
  indexes does not update column statistics. 
  
  Reorganizing an index, for
  
  
    example using ALTER INDEX … REORGANIZE does not update any statistics.
  


Question: Why would recalculating them separately cause an issue?

Might be because your sample rate is to low and the statistics is not accurate. Use FULLSCAN to see if that helps. It might also be that the Cardinality Estimation misinterprets the statistics (can happen in SQL Server 2014), if this is the case try trace flag 9481 to disable cardinal estimator in SQL Server 2014.

The StatisticsSample parameter of Ola Hallengrens script give you this: 


  Indicate, as a percentage, how much of a table is gathered when
  updating statistics. A value of 100 is equivalent to a full scan. If
  no value is specified, then SQL Server automatically computes the
  required sample.