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Great question! First, for me at least, memory palaces are asymmetric in effort, they can take a fair amount of effort to build (especially if you are trying to encode a lot of information), but retrieval is fairly constant (not instantaneous, but on the order of seconds.) As with anything, this gets better with practice.
And I don't want to mislead you! Memory palaces are fairly easy to get started with (like if you want to remember simple things, like a grocery list, you can plop groceries around a path in your house). The complexity in my techniques comes from developing a consistent visual language to encode things (like numbers with the Dominic System, or equations with the Extended Dominic System).
As for why I tried (and am still trying to do this!) there are a couple of reasons, First, I have been thinking about what it means to know something in the age of the internet. Like you pointed out, why not use the Internet as an external memory? By that definition, do I know all of Roman history?
Obviously not. Even if all the facts are there, they aren't in a format that allows for connections to be made. So I have a very similar idea as you do-- making knowledge internal allows us to develop intuition and connections.
I still think that this is broadly true. I used a memory palace to memorize the dates of office of American presidents and Canadian prime ministers, and I've noticed that it gives history a lot more context-- they act like meridian lines throughout history, reference points. My next project is to memorize English heads of states (they go back a lot further, so extending my temporal grid lines).
The second reason I'm trying this is more personal. I have always been amazed at human's mental capacity-- Greek poets reciting the Illiad by memory, Aleksandr Solzhenitsyn[1], who wrote a book in his mind using a rosary as a kind of word-abacus, the feats of calculation of human computers like Katherine Johnson, even reading about the Mentats in Dune was, and continues to be, inspiring to me.
If you look at the great strides we've made in understanding the human body in the last century, and our ability to train and condition strength, pack on muscle, whatever metric you want, you (or at least I) can't help but wonder if there is a similar revolution possible with our minds. (I know there is a bunch of studies that suggest we can't improve our working memory/IQ, but I take that more as a challenge).
The third reason is more... ahem... practical. People listen to you if they think you're smart. They will give you jobs and raises if they think you're smart. It also feels good when people think you're smart. And you can contribute more to a conversation if you don't have to say "Hold on, I need to google this." or "I don't know, but give me a day or so to think about it/google it."
Sorry for the word vomit
[1] https://nevalalee.wordpress.com/2016/07/11/solzhenitsyns-rosary/