A while back, I spent some time GMing the second edition of Paranoia. In second-edition Paranoia, players aren't supposed to know the rules (mine definitely didn't) so there was little temptation for them to describe their actions using rules terminology, and the mode of play we used occasionally involved taking players at their word and declaring that their stated actions resulted in hilarious consequences, so I got into the habit of using this style of play and later applied it to other systems, including Pathfinder. I later learned other ways of doing things, both from Pathfinder and from other games, so I'd say I've seen both sides of this issue; I've learned that enforcing fiction-first declarations can have a number of effects, some of which are beneficial, some of which aren't, and some of which are mixed blessings.
This style of play encourages players to engage with the setting fiction instead of the game mechanics, at least sometimes. Often, the intended purpose of requiring fiction-first declarations is to help players think in terms of the setting fiction, and it can do this. Sometimes. As I'll detail below, it doesn't always work, but there are other ways to encourage players to engage with the fiction, and there's nothing stopping you from using those as well.
This style of play may encourage players to learn specific formulaic phrasings that meet the GM's criteria for specific skills without actually using game terminology, but that's not usually a problem. Back in the day, I had a clear idea what each skill covered, and if a player's description of their actions matched the skill, I'd tell them to use that skill. The thing is, though, my players eventually learned where I drew the borders between skills, and learned to describe their actions in ways that unambiguously used the skills they were best at. Originally I felt frustrated by this, but another system, Mongoose Traveller, later taught me that it isn't a problem: If someone finds that one approach to problems works better for them personally than other approaches, they will tend to use that approach; that's not metagaming, it's what real people do, whether those people are your players or the characters they play. However, this potentially creates a few problems.
Some GMs may be jerks when using this style of play, and not necessarily deliberately. You know how some GMs like to twist the intent of every single wish spell the players get access to for optimum hilarity? Some GMs don't realise that they're doing that in other circumstances, or that the circumstances they're doing it in make it inappropriate. If a player declares "I draw my sword!" and the GM replies "The unarmed peasants cower and beg for mercy," the GM should absolutely listen when the player goes on to explain they meant to face the other way and act like they were defending the peasants rather than threatening them. Similarly, if a player describes an action with the intent that it should be carried out using a particular ability and the GM declares it should use a different one, the GM should let the player explain why the skill the GM picked isn't what the player would use. (Of course, listening to players doesn't mean agreeing with them; Some players will try to pull a fast one, and different games have different assumptions. Still, if a player's argument seems reasonable, it's generally best to assume good intent.) Also, players may have incomplete or flawed understanding of a situation, and may attempt actions that make sense in their heads but not in yours, so look out for situations where a player's interpretation of their actions differs from yours - you may be able to sort out some misunderstandings.
This style of play requires trust. There's a lot of different playstyles out there, and some of them don't play nice together; It's a sad truth that some groups have trust issues. Players who've been burnt (or who've heard worrying stories) may dislike "being forced give up control" of their character's actions to the GM, and if handled badly the entire exercise can make a GM look like he or she doesn't trust the players to play "properly" without it. I've never been good at overcoming trust issues (When I did that thing where one person leads another around blindfolded, I was slammed into a doorframe), so I've learned that it's better to prevent them from arising in the first place: Discussing in advance the intended purpose of a fiction-first declaration policy, and how various edge cases will be adjudicated, helps prevent animosity and feelings of betrayal further down the line. Whatever you decide in such a discussion should be considered part of your group's social contract.
This style of play makes the game harder for players who lack a full understanding of the fiction. The rules can be a rock to cling to, something players can use as a basis for making rational decisions in the absence of other information. And there's plenty of reasons to lack information; When players are tired they have trouble maintaining concentration, GMs can accidentally omit vital details, and new players joining an old campaign rarely have any idea what's going on. If you insist players describe their actions purely in terms of the fiction when they're not entirely clear on all the details of that fiction, they're likely to make some strange decisions. Of course, working to understand those decisions can give you opportunities to identify and explain the points that players have missed, so this is arguably both a bug and a feature.
This style of play requires the GM to keep more information in mind and on hand. Sometimes, a player will describe an action in a way that makes it ambiguous what skills they would use to accomplish it. (This happens all the time with knowledge skills in my D&D and Pathfinder campaigns, for some reason.) Ideally, I then judge what skill to use based on my knowledge of the player character's personality and abilities... But my memory is far from perfect, and my players have a much better understanding of their own characters than I do; I generally have to ask the player what their character's abilities are, and at that point I may as well just ask them what skill they'd think would make most sense. Unless I duplicate all my players' bookkeeping, there's no way to avoid this - and even if I keep careful notes, I'm not infallible; I'll occasionally make a bad call that a player (and a competent realistic player character) would not.
Tl;dr: In my experience, enforcing fiction-first declarations is a mixed bag. There are benefits to it, as well as costs; In most of my campaigns I encourage fiction-first engagement but don't actually enforce it, and that seems to work for me.
(Oh, and I've heard tell of groups that treat campaigns as a series of tactical combat puzzles and engage with these puzzles almost entirely in terms of mechanics. I have no experience with this play style, but from what I've heard, it does not mix well with fiction-first declarations; Particularly in mechanically-complex systems, the effort of resolving extremely complex mechanical situations purely in terms of the narrative can be a massive cognitive burden that drastically slows down play.)