In episode 1 of Loki, Glorious Purpose, Loki watches a TVA recording of his life, shown in the third person:
Why is it shown as if someone is looking at him, rather than from his perspective?
Science Fiction & Fantasy Stack Exchange is a question and answer site for science fiction and fantasy enthusiasts. It only takes a minute to sign up.Sign up to join this community
Presumably, it's because that footage is not a recording of Loki's memories. It's some form of 'camera' footage, quite possibly captured by He Who Remains himself.
Bear in mind that the main Loki variant played by Tom Hiddlestone on the show was plucked from the year 2012, during the events of the first Avengers movie, so he didn't have any memories of the events of Thor: The Dark World (set in 2013), Thor: Ragnarok (set in 2017), or Avengers: Infinity War (set in 2018).
As far as we know, the TVA never captured the Loki from the later movies who did experience those events, and therefore did possess those memories. And even if they did capture him, there's been no indication that the TVA are capable of directly viewing people's memories from a first-person perspective.
He Who Remains also made it clear that he was somehow capable of observing events occurring within the Sacred Timeline, in a way that the rest of the TVA could not. How he managed this is unclear, but there's no suggestion that he was seeing things from any perspective other than his own.
HE WHO REMAINS: Don't you wonder how I'm able to get out of the way just before you kill me?
SYLVIE: No. It's because of that little TemPad you have there.
HE WHO REMAINS: Right. But how do I already have it loaded up with everything I need to know to keep from being killed by you two? It's easy. I know it all. And I've seen it all. Everything you guys did on Lamentis, I saw. All the stuff the TVA didn't know about, I knew. All the scheming, all the... talking. That little look by the lake. Quite sentimental, very touching stuff, by the way.
Loki - S01E06 - "For All Time. Always."
This is never mentioned in the show, but I considered it implied that, as 'stewards of the timeline' who have the ability to monitor all of existence for variations/nexus events, the TVA also has a "video output" option to choose from when reviewing said timeline.
Out of Universe, they didn't have footage of all the events from Loki's perspective, so I assume they just re-used footage they did have for practical reasons.
I would venture to say that the technical answer is that the cast/crew are using footage from previous films and have to use the framing and cameras that the originals were shot in.
From an in-universe perspective, consider that the TVA are meant to be showing Loki an impartial and unbiased view of his life. To show things from his perspective might be to lose context or to paint things with his own personal bias.