While I wouldn't discount the occasional (or even often) slipup by saying "arrived in the X system", often what you'll hear is " approaching the X system".
That is, the announcement is taking into consideration that they are well enough away from the destination for it to matter.
Why would they announce it ahead of time instead of just doing it?
Consider another,
similar paradox: Enterprise is in pursuit of another ship at high warp when the helm says, "Captain, they've dropped out of warp!"
The captain then says, "take us out of warp, lieutenant."
In the time to relay the condition and give the order, Enterprise should be nowhere near the ship they were pursuing.
To reconcile this, it's important to understand that creating and destroying a warp field is not instantaneous: this is usually glossed over in the episodes, but it's illustrated to dramatic effect at the beginning of Star Trek:
The Motion Picture , where the bridge officers spend a good 5-10 minutes getting anxious while bringing the Enterprise up to Warp 1 after a major refit.
When the helm prompts the captain that they're approaching a star system or that the ship they're pursuing is dropping out of warp, they're taking into account the time and relative effort it takes to drop out of warp and asking the captain if they can proceed.
Why ask?
Chain of command.
The helm gets his or her information on a need-to-know basis: the captain could've received new information or wants to make a tactical decision that would contradict the obvious next step (drop out of warp).
So the helm asks for confirmation of the new task once he's completed his previous one (navigating the ship at warp).
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IMHO, this is mostly because in a lot of space sci-fi - Star Trek included - space ships are usually modeled after regular wet Navy ships.
Navy ships move slowly enough that this formalized real-time chain of command is fully adequate as far as human reaction speed.
Unfortunately, Star Trek is a Shakespearean Theater in sci-fi setting.
So it didn't bother getting many details right scientifically - including this one - unlike a lot of good hard sci-fi, where the authors actually put in a bit of thought, realized this would be a problem, and switched to a more appropriate captain pre-selects parameters (e.g. how close to system to drop out of hyper-speed), crew enters parameters into the system, the system exits hyper-speed at pre-programmed point .
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The comparison to a naval "look and feel" is quite apt.
It's partly a question of aesthetic preference that overrides the science.
But consider.
Even in modern naval maneuvers, and especially in naval (or really any) aviation where timing is much tighter, there is not always time to actually process decisions in the way that the formal chain of command would theoretically imply.
Often in these real-world cases what will happen is that everyone already knows in advance what needs to be done and the crew is already effectively executing the command as, or even before, the captain issues it.
Even a routine trip from one planet to another probably involves an enormously complex flight plan (or "warp plan") that is meticulously worked out in exhaustive detail long before the ship makes planetfall.
If the ship is on course and cruising according to plan, the captain may be formally informed of their impending arrival long after the preliminaries of the de-warp procedure have already begun.
Presumably the captain knows this already and is perfectly aware of how the journey has been progressing.
Everyone is sitting there calmly as the captain almost off-handedly gives the (formal) order because no one is in the slightest bit surprised, and indeed as far as the below-deck crew is concerned the maneuver may already be in progress by the time they actually hear him sign off on it.
This is one reason, incidentally, why quality of crew training and the fine line between discipline and initiative are so important.
You want combat officers who know that if you are following a pursuit plan and the target starts coming out of warp, that you should start coming out of warp too, even if the order hasn't technically been confirmed yet from the captain's chair.
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It can have a very plain explanation -- warp drive works by putting a ship in a spacetime bubble which may be sliding through normal space at FTL, but maybe not necessarily has to .
In other words both warp and impulse would use spacetime folding, and going out of warp would only mean switching from faster to more maneuverable drive.
This has some justifications in some mentions that impulse may also be FTL, and is reasonable because conventional flight with near c is also quite problematic -- the energies are so large that a collision with a grain of salt could easily destroy the ship.
Anyway in this scenario the ship may just slow down to 0.1-0.01c, which is quite a reasonable speed to cross a solar system and give the captain a chance to pick a planet and orbit.
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I think there are a number of reasons behind this.
Primarily, it is a plot device - it says "we are arriving at x".
The captain needs to be seen to be driving the ship - as per DVKs comment - and so has to make a response.
In reality, the process of bringing the ship down to sub-light speed at this point, and it is merely a confirmation that this is still the right thing to do.
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Often when they arrive in a star system from warp, someone will say something like, "Captain, we've arrived in the such-and-such system." And he will give the order to take the ship out of warp. Now, at warp speed, if he responds too slowly, even by a fraction of a second, why doesn't the ship end up light years away?
Why does Picard need to give verbal orders to bring the ship out of warp?
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Primarily, it is a plot device - it says "we are arriving at x". However, while he may sometimes say "arrived in the X system", often what you'll hear is " approaching  the X system". This is mostly because in a lot of space sci-fi, space  ships  are usually modeled after regular Navy ships. There are as number of theories as to why the captain has to give the order such as; space ships being designed with navy like hierarchy, the idea that this is just a plot device, warp drive not having to function at faster than light speed etc. However these are all theories and there is no canon explanation for it.
In most cases the pilot states that they are "Approaching" location X, which in theory gives the captain plenty of time to make the order. The space ships hierarchy resembles that of what is found in the navy and the writers of the Star Trek universe specifically designed it that way regardless as to how realistic this is to the situation. Much like the hierarchy in the navy they have to wait for captains orders as they may decide to make a decision outside of the obvious for tactical reasons. Warp drive works at faster than light speed, but it can probanly funking at less. Primarily it is a plot device used to move the story forward.