YANBU.
I suppose we 'gentle parent' although I dislike the term and tend to explain, mostly impose natural consequences, encourage discussions and questions and keep firm boundaries but our number one rule in our house is that we are a family and we respect each other's feelings and emotions as much as our own.
I was raised in a very corporal punishment household as was DH, lots of shouting, smacking, fear, high expectations and it wasn't a cycle I wanted to continue. It's very much been a learning curve as I am naturally very quick to anger at aggression, however, as the children have gotten older we have learnt together.
Our style of parenting is very similar to yours. We outline boundaries and expectations and if they are crossed/not met then we respond accordingly I think it's called authorative? Whereas your friend's parenting is permissive, there doesn't seem to be any discipline. Kids need age appropriate consequences as much as they need understanding.
For my children, shouting doesn't work. It escalates the confrontation and merely shows them when we want someone to listen we have to shout. That doesn't mean I haven't absolutely lost my shit, quite the contrary, but afterwards I always acknowledge and apologise and I expect the same off them.
Whilst I am very aware of their emotions/moods, that does not make them unaccountable for their behaviour. If they are unkind/shout then I encourage them to calm down by mirroring deep breathing, doing self squeeze, stress balls etc but reiterating shouting is unacceptable.
Once they have calmed down, we deal with their problem, explain why their behaviour wasn't a good choice and we can lose privileges.
Last weekend DS woke up in a stonker of a mood, I was patient, patient, patient, reiterated boundaries and warned if his behaviour continued he would not be going to the big park, asked if he wanted to talk about anything, attempted to redirect etc but it didn't work. He was overly tired but that didn't excuse his behaviour.
The final straw was he ripped his sister's art work up so after comforting her. I went back to DS then explained very calmly Mummy had told him repeatedly that he was not making good choices and he would not be going to the park. He had upset his sister and ruined something she had worked very hard on. She had been very kind to him all day and that behaviour was not being reciprocated, he had now lost the privilege of going to the park.
He cried and he kicked off, shouted, threw himself around but he had been told. When DH came home from work he stayed with DS and I took DD to the park. We came home and he'd fallen asleep on the settee (very out of character for him) but woke up in a better mood. He apologised to DD, drew her a new picture and apologised for his behaviour. We talked afterwards about why Mummy hadn't taken him and why, he understood. We also talked about how he can express himself when he's frustrated and what is acceptable.
In your friend's shoes I would have given a very calm. 'That has really upset me. I love you very much and I've tried very hard to make your day special for you. How would you feel if I said that to you?' Birthday boy he would have to have that discussion, apologise and would have consequences.
We are raising our kids into the adults they are going to become and if you let them off the hook constantly because they're tired they will grow up thinking that's acceptable behaviour. Allowances should be given but not endless free passes.