I've never had to deal with someone or a pet dying like that, to be there in their final moments so I can only imagine what that must feel like and the impact it must have on you. One of the reasons I obsess on ghosts is because I long to see the people and pets that I've lost and never got to say goodbye to again properly as well as the people who I will inevitably lose over time too.
I've heard of that tbh, you focus on making the dying person or pet happy as possible in their final moments, try to give them the best life you can when you know their time is running out and even if its happening soon ofc.
I used to want to volunteer in hospice work and I heard this is essential. You try to make them happy in their final moments, because they will be terrified, they will be upset and such naturally, so you have to comfort them, make the dying patients try to not pass away in torment, fear, agony or whatever.
I can understand that notion of saving your tears for later. I've seen parents talk about knowing when their children were gonna die, children with cancer or so and trying to hide their pain, their sadness so they can focus on giving their child the best remaining months, weeks or whatever whilst they still have the chance to. Making the most of the extremely limited time life they have with them until they are gone permanently.