My dad bought our PS2 from some guy on the internet and it came with a binder full of games. I didn't know what Ico was the first time I started it up, from the intro I expected some kind of medival hack and slash for a while. Once I started playing it I liked it though, the movement reminded me of Flashback and Prince of Persia (also Abe's Oddysee), games I had fond memories of from when I was younger. The combat was simple and kind of terrible, but it also made sense to me that it wasn't the focus of the game.
Going back and seeing the design of the puzzles and how you are supposed to figure some things out in both Ico and Shadow of the Colossus I would agree that they are not intuitive, but at the time I was used to that kind of design from a lot of other games I had played growing up. I remember liking Ico, being invested in what story there was but being afraid that I would get stuck somewhere and wouldn't be able to finish the game and see the ending. You see, at the time that was still a fairly common experience with games for me (I have to this day never finished the final level of Flashback, not to bring up countless NES games), but today the idea that you wouldn't be able to finish a game for any reason other than that you didn't invest enough time in it is very unusual. This in part kept pushing me forward and it didn't annoy me as much as it challenged me in a way I expected.
I don't think neither Shadow of the Colossus, nor ICO has good intuitive design by modern standards but for me personally it is not a problem as much as a throwback. If The Last Guardian comes out and has the same kind of design I'll play it and enjoy it, if anything it makes a short, unique game last longer. I won't defend the games combat other than to say that making it deeper system with blocks, dodge rolls and combos would have clashed with the sort of character you were playing. It still could have felt less clunky I'd imagine.
I will however defend the movement and platforming, it's weighty and systems-heavy, like the games it took inspiration from. It's different but not bad.