The big thing missing from this discussion is executive producers.
A producer can be anyone - there have been people credited as producers who literally weren't aware the film or show was being made.
But an executive producer means there's some actual involvement with the show - now that exact level of involvement varies, but in a lot of cases it means both organising funding, and to a greater or lesser extent influencing how that funding is actually used. This is increasingly true, as I understand it.
With TV shows, the showrunner or showrunners are typically executive producers as well as writers, and more rarely, directors. And the buck, ultimately, is generally going to stop with them, as to whether a show looks cheap.
With movies, the director may or may not also be an EP, but depending on the movie, the studio, whether it's part of a franchise/existing IP or original, and whether the director is a director famous and followed in their own right or not all influence who actually gets to determine how the money is spent (among other things). Usually the director is basically to blame (the cinematographer and DP answer to the director, as do others, you can't blame them, generally speaking), or and where the buck stops.
It's worth noting that even with movies, EPs have often interfered with movies, to the point of really annoying a lot of directors, and in the view of certain directors "ruining" a number of movies. And they've certainly contributed to movies looking "cheaper". Kevin Feige, for example, I know people love him, but he insisted on putting some really awful colour grading on a bunch of MCU movies, often over some degree of objection (mild to strong) from the actual director(s) of the movie in question, which to me, unquestionably made some of them look both cheaper and more artificial. And that's a mild example - editing can really impact a movie and a number of EPs have decided they were going to do the final edit, not the director.
But to return to the first question, with a movie like Doom, the reason it looks cheap is probably going to be a combination of director and budget. $60m was a decent amount for "a movie" in 2005, but it wasn't crazy money.
Compare it to some other 2005 films:
Harry Potter and the Goblet of Fire - $150m
King Kong - $207m
War of the Worlds - $132m
Batman Begin - $150m
Mr and Mrs Smith - $110m
$60m is at the very low end for a 2005 sci-fi action movie. So I think the main reason Doom looks cheap is that they were pushing the budget a bit to do what they did.
Now interestingly Sin City that year did manage to fairly fantastic on $40m, but shooting B&W and going for a highly artificial look clearly helped it a lot.
Who is to blame there? The director I think in the end choose to make the decision that lead to the Doom movie looking how it did. He could have toned things down - he have got the script changed to make it less flashy and more tense, which is pretty much always cheaper - he could have picked ways of shooting that didn't make it look cheap.
With something like Wheel of Time the buck stops with the showrunner who decided to make it so location-heavy. Location shooting is incredibly expensive. The trouble is it's a bit damned if you do, damned if you don't. You want to shoot a fantasy show in HD and make it look convincing? You're going to need a lot of $$$ and you're going to need a ton of location shooting, a ton of CGI, or both. The large and diverse array of costumes, which needed to be good enough to work on HD TV probably didn't help (and some were, but some really weren't).
Wheel of Time was also not as overbudgeted as people seem to think - it essentially got hit by the COVID tax.
There is another culprit in making a lot of stuff look cheap, though, who we have all so far overlooked.
Motion Smoothing.
No a strangely named Hollywood exec, but a feature most people's TVs, particularly 120hz TVs, have and which with very many TVs defaults to on, and even if turned off, likes to find exotic ways to turn itself back on. Motion smoothing will make anything look cheap. Indeed it is particularly good at making everything look like it's on a set (which to be fair, it usually is) and any kind of framerate above 30 seems to be pretty good at making props look cheap (you can fight this, but it requires special attention and using very realistic-looking props - more realistic than most movies/shows us).
Several times I've come across someone saying "X looks cheap" or whatever, and it turns out that motion smoothing had been engaged when they watched it. A friend of mine did this a few years back - turned down the lights, switched the TV to the pre-configured "movie" mode, which engaged a bunch of stuff, but which inexplicably or possibly through the work of the devil or demiurge or similar, turned on motion smoothing. I spotted it pretty quickly, but he was very sure it wasn't on because he'd turned it off. I have another friend who actively turns on motion smoothing - I don't take his advice re: anything about aesthetics or movies/TV!
As a bonus, many TVs rename motion smoothing in order to make it sound like it's some kind of exotic cool feature you want on, which is pretty evil imho.
