This is a personal choice.
There should be no guilt or shame either way.
Many people that think they can’t bear to be in the room with their pet during the process go on to later regret that decision. This is an extremely emotional decision and moment, and you should expect great sadness and tears. Your discomfort should not govern your decision of whether or not to be present with your pet. Being present will help your peace of mind after the fact because you will know that they passed on peacefully.
However, this is such an individual decision, and there should be no guilt either way.
If you really can’t be present, let that be okay. Many people find this too distressing and can’t handle the process. If you are in this category, there should be no guilt in leaving the room. Whether you chose to be present or not, you are still caring for and loving them by giving a peaceful release from pain. Your pet knows that you love them, that you take care of them, and that you are not abandoning them. They trust you to make this decision for them.
Should you let your children be present during the procedure?
This is entirely up to their individual developmental level; can they emotionally handle it or will it be too traumatic? If they are ready and willing to be present, make sure that they know what to expect during the process. Regardless, your children should at least know that this is happening and be given the chance to say goodbye.