The difference between a "gravel bike" and an "endurance road bike" is pretty blurry especially in your price range. I'm sure any of those will be fine, if you find yourself flatting you can get armored or bigger tires. You can find a way to put fenders on any of them.
But any bike like that with a drop bar is made for covering lots of miles quickly for hours, with padded shorts and the whole bit, turning every muscle twitch into forward motion and keeping you just barely comfortable enough to not give up. That doesn't sound like your ride, which is going to take maybe 20 minutes.. In fact a 90's MTB is a popular style bike for commuting. Is yours worn out or are you just tired of it, or what specifically don't you like about the ride? They take standard parts, nothing on it is too obsolete to service. A road bike is obvs faster but you can speed up your MTB with some slicks, and over 3 miles the difference in speed just really doesn't matter.
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Genesis 49:16-17