+/- 1 day should be no issue with agent. This is within policy for schedule changes. There was a change that OTA can no longer make this change. However they just need call DL support if passenger wants, as it is still within policy.
They had issues with OTAs going wild with new flights. Ie booking passengers with 5 minute changes to next day, not correctly reissuing tickets, etc... They had to impose some limits.