Hi Beliver,
Tantrums are difficult! The child's emotions are raging so hard calming them can be challenging. It's hard watching your child suffer so, but I think at times tantrums can be used as a tool to gain control, particularly when the child gets their way afterwards.
My firstborn was subject to having tantrums. He would cry until he turned blue and sometimes pass out, which totally freaked me out. That last time he did that I put his face under a faucet and turned on the water. He never had another tantrum. Lol the 'abuse police' will probably have something to say about that but the results speak for themselves.
My grandson had a couple tantrums with me. I forced him to toe the line while having the episode, and when he didn't comply I would begin taking away privileges. He could continue crying if he wanted but he had to obey me and accomplish the task too. I also believe in spanking (though I don't spank my grandchild), but I do not believe spanking a child while they're in the middle of an emotional rage could accomplish anything meaningful. Just my opinion.
Since they are five, they can reason now. If I was giving advice to my daughter, I would tell her to speak to my grandchild lovingly and do not judge them for being overly emotional, but do not acquiesce to it. Also never argue with them about carrying out their responsibilities; put your shoes on, take your toys to your room, share with your siblings (that's a fun one!), brush your teeth... These things are non-negotiable. Of course thee are always concessions but don't make them in the middle of a tantrum. YOU are in charge right now. Be careful not to confuse the child on this issue by waffling during a tantrum. Be firm. Children need boundaries, especially during times when they are out of control. Gently bring them back into control by keeping those boundaries ever do lovingly. Set boundaries securely and don't negotiate on the important stuff. Know you are giving a gift to your child by being the one in control.