Why shouldn't I increase SQL Server's timeout globally everywhere?
So we decided to increase database timeout for queries globally. But I
  have this feeling that this is wrong. Yet I'm not able to bring
  forward any reasons. Is it bad to increase timeout everywhere? Why
  shouldn't we do this?


The default client API query timeout of 30 seconds is appropriate for a typical OLTP workload, albeit arbitrary.  The value should be set high enough to allow queries on a reasonably healthy system to complete successfully yet low enough so that queries timeout when the rather than wait indefinitely when things aren't well, such as in a long-term blocking scenario or due to a bad execution plan.  There is no single timeout value that is appropriate for all queries unless the system is a 100% OLTP workload or 100% reporting.

It would be better to first address the root cause of longer-running queries before increasing the timeout value.  Perform query and index tuning, ensure hardware is adequately sized and configured properly for your workload, and make sure EF data access is optimal.  These practices will make the most efficient use of available resources, reduce compute costs, and provide the best end user experience.  

If the amount of data required by queries doesn't increase with table size yet queries run longer as the number of rows in tables increase, this suggests scans. Query and tuning would be a good first step, assuming SQL Server memory is appropriate for the database size.