I find it fascinating that people, myself included, get nervous about speaking in public. Logically, I and most people can talk at the dinner table in front of my family. Most people can be at a party with friends and announce something "Time for Cake!", "Hey! Group Photo!". Most people can be in group of 4-8 friends and say something to all of them "Hey, how about pizza?" etc.... Most people can be in team meeting and state what they are working on, what problems they're having, and give their opinion.

In a certain sense, public speaking is no different than those situation. Even better, the people you're in front of you don't really care about, where as you do care about your family and friends. In other words, it should be that embarrassing yourself in front of friends and family "should" be the worse thing since you will have to face them again where as you won't have to face these strangers you're speaking in front of.

And yet, even though I've given ~10 public talks I still feel those butterflies, at least until the talk actually starts. Once it starts, for me, they go away. That assumes I know my topic. I'd say a best person talk at a wedding is more nerve wracking, for me, than a talk about anything I'm working on. I can talk about what I'm working on easily because I know it inside and out. Sure I need to practice my talk but if someone came up to me in a hall and asked me about my work I could talk their ear off with no practice. Conversely, the wedding talk topic is not a topic I know inside and out, hence it's harder, at least for me.

I know it was certainly nerve racking the first time I had to give a public talk at school. I wonder if the teacher had asked in private, something like

Teacher: Are you nervous about your talk?

Student: Super nervous

Teacher: Do you have a favorite video game?

Student: Yea, Minecraft

Teacher: What do you love about it.

Student: (with teach prompting, goes on for 5-10 minutes what things they love about Minecraft, things they made, experiences they had...)

Teacher: You just gave a public talk to an audience of 1. Just organize and practice a little and you'll do great!

I don't know if that would help but I'd like to think it would.

With family and friends you can just be yourself[1] though. If they watch the recording after 5 or 10 years, they will probably look at it affectionately, at most with some light teasing.

They're not going to, for example, start harassing you online just because at some point during your presentation you said the word "blacklist" (or other words that are nowadays are considered worse than they were back then).

So I'd say part of the reason of why it's easy with family/friends but stressful with strangers, is subconscious fear of this kind of judgement where you don't know how this large amount of people will react, and fear that something you say or something you do will cause a response way more negative than you expect. Sure, you can imagine some possible negative responses, but because you don't know them personally, you don't know what's their "upper bound" for a negative response (how far they will go).

And once you start talking the nerves go away because, once you start, running away suddenly becomes a worse choice (easier to get a negative response) compared to finishing the talk.

That would be my guess.

[1]: Some might cross lines, but those should be the exception, and you probably wouldn't be relaxed with them anyway.