Well, I don't necessarily mean it as a bad thing when I call Chief shallow. I feel like Halo is a universe where all the characters are soldiers and everyone has been at war constantly throughout all of recorded history, and even among that crowd Chief sticks out to the people around him as being single-mindedly obsessed with fighting bad guys. When I think about a character with some depth, I think about whether I can imagine that character on an off day, when they aren't at their day job. I can imagine, say, Nathan Drake when he isn't on an adventure. I can imagine Old Lara Croft when she's just relaxing around the manor, but I can't imagine Nu Lara Croft that way. I can imagine the Solid Snake of MGS1 and 2 as a regular man, but I can't do it for MGS4's Snake; I guess it's a quality of writing and characterization that I can't really put into words, but I know it when I see it.
When it comes to Master Chief, I absolutely can't imagine him doing anything but shooting guys, but, like you said, the canon of Halo more or less categorically states that he doesn't do anything but shoot guys. He basically doesn't even get to take a shower from the opening of Halo 1 to the end of Halo 4, right? And then at the beginning of 5 his buds mention under their breath that he's been running missions non-stop since the last game. And I think that's totally fine. That's what Halo is; the fiction is all about Chief being a manmade supersoldier who doesn't stop, and I think it'd be a mistake to start depicting Chief sitting on the space couch watching football or whatever. He doesn't need to be a deep character. You mention Doomguy, and I'd argue that 2016's Doomguy carved out more of a personality for himself from that miracle-energy-source destroying scene in Doom 2016 alone than Chief has in like five games (I will give him points for jumping through space all those times, though). Anyway, I'm still fine with him being shallow.
It is why the emotional Cortana/Chief stuff in 4 just didn't work for me at all, though. Chief isn't a character I can imagine being sad about things, or forming any kind of emotional bond beyond a Chris Redfield RE5-style "she's my partner" kind of thing, which I'm fine with, but it really puts a limit on the kinds of stories you can tell with the guy. On a pure gut reaction level, I just didn't buy Halo 4's relationship between the green deathbot and the blue sexbot. This shot of him just standing there at the end really put the capper on the whole thing. I just don't know what they were thinking with this. He just stands there. Have him drop his gun. Put his hands on his head or something. Drop to his knees. Melodramatic, sure, but it'd be something. He doesn't have a face, he doesn't move, he doesn't express any emotion vocally, he just stands there.
To be fair, I think Halo 5's story worked a lot better. I found Chief's voice actor a lot more convincing, and I think the premise of "we're trying to stop Cortana being evil" worked a lot better for Chief's limitations than "she's dying because lore and the emotional crux rests on Chief being sad". And 5 had some other characters and subplots going, too, so Chief didn't have to pull all that weight himself. And also Cortana had some clothes on this time
Anyway, I think that's pretty much how I feel about these Halos. I don't want to say they shouldn't ever try anything more with Chief's character, I just... hated it when they did it. They could always try again. I'd like to see it. I'm still looking forward to Halo Infinite.