The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally.

First, there is no such thing as training to be a number theorist. There are many different kinds of number theorists, and very few of him are comfortable with all four of the works you mention here (Davenport, Vaughan, Deligne-Weil II, Tate' thesis). Very few analytic number theorists understand the proof of Weil II (though a lot more of them know something about how to use it). Very few algebraic number theorists are comfortable with all the standard argument in multiplictive number theory and the circle method (though a lot more know the key results about $L$-functions). Of course the division into analytic and algebraic is alredy too coarse and simple. What you have is a lot of different number theorists with distinct but overlapping areas of knowledge.

Analytic number theory specifically is one of the areas of math famous for requiring relatively little knowledge (at least, when compared to other areas of math). If you like the stuff you read in Davenport and Vaughan, you're in luck! You may be a lot closer to the frontiers of research than you think. As for how exactly to get there, I agree with Timothy Chow that your adviser is the best person to figure this out.

As to this phenomenon:

When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about.

Their knowledge may be less than you think. Or more precisely they know a broad overview of what the idea in a given field are and how they are used, but not the details. This might match the questions that someone with less experience in the area would ask them, but not be sufficient to write a good research paper in that field.

However, it is by no means a parlor trick. This type of knowledge is very important because it suggests what research areas might be relevant for a given problem and thus who to talk to, what to read, etc. But it's not obtained from reading books! Probably the best way to attain this level of knowledge is attending seminar talks (and listening carefully, not being afraid to ask stupid questions, thinking about what the speaker is saying during and after the talk...)

I think in general, a recipe for success on a particular problem or research sub-sub-area is to know (1) everything, or as much as possible, about the techniques that have been used to attack this problem before and (2) one relevant thing that hasn't been used to attack the problem before. The point being that you only need one new idea to make progress, but you will likely have to combine it with all or many of the previous ideas.

So if you know which problem, or type of problem, you want to work on, you should learn diligently the topics of obvious relevance to that problem. For topics of unclear relevance, you do not need to learn everything to their fullest extent, as long as you do not completely abandon them - again, you only really need one new idea. Even (2) is not strictly necessary - plenty of progress has been made by applying the existing methods with a more clever strategy.

But if you have a natural inclination to read and learn everything, you will probably find success as a mathematician by knowing at least a few things that your competitors don't. Focus on what seems relevant to your areas of greatest focus and ideally what seems fun and intereting as well. But there's no need to drive yourself insane.

The other answers have some good general advice. Let me try to say something that is specific to the topics of analytic number theory, and number theory generally.

First, there is no such thing as training to be a number theorist. There are many different kinds of number theorists, and very few of him are comfortable with all four of the works you mention here (Davenport, Vaughan, Deligne-Weil II, Tate' thesis). Very few analytic number theorists understand the proof of Weil II (though a lot more of them know something about how to use it). Very few algebraic number theorists are comfortable with all the standard argument in multiplicative number theory and the circle method (though a lot more know the key results about $L$-functions). Of course the division into analytic and algebraic is already too coarse and simple. What you have is a lot of different number theorists with distinct but overlapping areas of knowledge.

Analytic number theory specifically is one of the areas of Math famous for requiring relatively little knowledge (at least, when compared to other areas of Math ). If you like the stuff you read in Davenport and Vaughan, you're in luck! You may be a lot closer to the frontiers of research than you think. As for how exactly to get there, I agree with Timothy Chow that your adviser is the best person to figure this out.

As to this phenomenon:

When I look at some of my professors or other researchers I have interacted with, I notice that they may be publishing in one or two areas, but are extremely knowledgeable in pretty much everything I ask them about.

Their knowledge may be less than you think. Or more precisely they know a broad overview of what the idea in a given field are and how they are used, but not the details. This might match the questions that someone with less experience in the area would ask them, but not be sufficient to write a good research paper in that field.

However, it is by no means a parlor trick. This type of knowledge is very important because it suggests what research areas might be relevant for a given problem and thus who to talk to, what to read, etc. But it's not obtained from reading books! Probably the best way to attain this level of knowledge is attending seminar talks (and listening carefully, not being afraid to ask stupid questions, thinking about what the speaker is saying during and after the talk...)

I think in general, a recipe for success on a particular problem or research sub-sub-area is to know (1) everything, or as much as possible, about the techniques that have been used to attack this problem before and (2) one relevant thing that hasn't been used to attack the problem before. The point being that you only need one new idea to make progress, but you will likely have to combine it with all or many of the previous ideas.

So if you know which problem, or type of problem, you want to work on, you should learn diligently the topics of obvious relevance to that problem. For topics of unclear relevance, you do not need to learn everything to their fullest extent, as long as you do not completely abandon them - again, you only really need one new idea. Even (2) is not strictly necessary - plenty of progress has been made by applying the existing methods with a more clever strategy.

But if you have a natural inclination to read and learn everything, you will probably find success as a mathematician by knowing at least a few things that your competitors don't. Focus on what seems relevant to your areas of greatest focus and ideally what seems fun and interesting as well. But there's no need to drive yourself insane.