EDIT: since you are of a normal weight and work out regularly, I trust that you know how to draw the line between exertion and discomfort, which is a blurry line that only you can draw. So I'm going to assume that the discomfort you feel is something you have deemed "not normal", especially considering the relatively short amount of time you spend in the drops.

I think your bike fitter is going in a good direction. There is no such thing as a "best" position, only optimal for certain situations subject to constraints. Conventional bike wisdom states that while mashing hard in the drops, you should barely need to support any weight with your arms. Inversely, if you are *not* mashing hard in the drops (ie you are a beginner), your arms would have to support more weight. This weight has to go *somewhere*. Saddle fore-aft position allows you to determine what the distribution of your upper body weight is between your arms and your butt. The further forward you are, the more weight you have on your arms (think TT or tri bikes, which have elbow pads). The further back you are, the more weight you have on your seat (think MTB riders, who sometimes need to take all weight off their front wheel). You can also dynamically adjust this weight distribution by bending (getting into the drops vs hoods or tops). Using a combination of these, you need to figure out a saddle position and bar position that works best given the distance, terrain, and speed you are riding. Since you are riding small distances, I will assume you are a beginner who doesn't put down monster torque. This would mean you should set the saddle back (*very* slightly) more than you would if you were super strong. Your dynamic range of weighting should also not be between "aggressive" on the hoods and "really freaking aggressive" in the drops like you see for pros. Instead, you should be more like "chill" on the hoods and "kinda aggressive" in the drops.

One benchmark you can use to see if you are in a relatively aggressive position is the Knee Over Pedal Spindle (KOPS) method. It is by no means a hard and fast rule, but try and see (perhaps using a weight on a string) where the bottom of your kneecap lies relative to your pedal spindle when in the "power" 3 o clock position. Depending on your proportions, road bikers like to keep it behind the pedal spindle by about 1 or 2 cm. If you're climbing a lot you may want to be *slightly* forward so that your relative position to the bottom bracket is efficient when climbing, though this would put more weight on your arms when riding flat roads. If you as a beginner are exactly at KOPS or even in front, I would be suspicious of the fit. You may feel that your quads have to do much more work than your hamstrings or glutes.

As you get stronger, you will put down more torque and need more weight on the pedals to counter that torque. This is where a more aggressive position comes in handy. This isn't any information that is specific to you, but it's information that I've personally found useful in my journey as a beginner.