Grandparents raised children in a different era, often with firmer potty rules and less awareness of withholding. When they see a child cross-legged and clenching, the instinct to 'just get them on the toilet' is strong, and they may believe you are being too soft.
Nannies and babysitters usually have the opposite problem: they want to avoid any upset on their watch, so they offer sweets, screens or a diaper the moment a child protests. That can be fine, but as a bargaining chip it teaches the child that poop is a transaction.
A child who is holding reads every adult for safety. If pooping is calm with one person and a battle with another, the child does not split the difference; they hold more, because the toilet is now unpredictable. That is why one plan matters more than the best plan.
The practical obstacle is rarely disagreement about goals. Everyone wants the child to poop comfortably. It is that nobody has been given the same short explanation and the same words. Writing them down changes the conversation from opinion to plan.