Is my old Weller station actually bad, or am I just blaming the tool?

Most of the time when older stations struggle with specific connections, the issue isn't the heater itself, it's the interface between the tip and the joint. Since you mentioned the tips have pitting and you aren't tinning them between uses, that immediately suggests a lack of thermal mass transfer. Pitting ruins the surface tension and prevents proper wetting, meaning the solder isn't flowing correctly onto the pad or component, forcing you to add more heat and apply more force, which is the wrong way to do it. When dealing with smaller or thinner pads, the amount of heat you need is very specific and minimal. If you are applying excess heat, you're just heating the solder and components past thier tolerance. A well-maintained older iron can do this perfectly fine. You need to focus on keeping the tip perfectly tinned at all times, which means a fresh coat of solder on the tip before every session. This coating is what conducts the heat efficiently into the work.

If the tips are heavily pitted, even a great station can't compensate because the surface area for heat conduction is compromised. You need to replace the old tips with new, high-quality ones that match your iron's power output, even if it's a fixed-temp model. You can use a cleaner to remove the oxidation, but a fresh tip is usually the best immediate solution. The heat you are feeling pull from the pad is likely because your current technique is forcing the solder onto the component rather than the tip properly interfacing with the joint.

Before you spend money on a new power supply or digital temprature controller, try buying a few decent replacement tips and strictly following the proper tinning and cleaning procedure. This solves the vast majority of 'hot/cold joint' problems in the hobby. A good, practiced hand with clean tools will almost always outperform an expensive station used incorrectly.

I really agree with what was said about the interface being the biggest problem here. It totally makes sense that the pitting and poor tinning are the bottleneck for heat transfer. Last year, I went through a similar process and decided to test a couple of diferent flux blends with my replacement tips. Even with a brand new tip, I found that using a high-quality flux made the solder flow onto the pad much faster than the basic stuff I ussually bought, wich really sped up the process.

I think the focus needs to shift slightly from the iron itself to the whole soldering sequence. Before you swap hardware, try tightening up your procedure. When I am working on small through hole components, I keep the temperature set around 360 degrees C, and I ensure the solder touches both the pad and the tip at the exact same time for a minimum of two seconds. The speed of that heat transfer is often more critical than the absolute power of the station. If you are touching the joint and tip separately, you are fighting the heat transfer. Try focusing on that simultaneous contact and the dwell time first.

I guess I totally agree that the preparation is everythign, not just the iron itself. I've been trying to get better at it too, mostly when I'm working on these larger PC components and bigger holes. My concern is that if I'm focusing so much on the tip not scorching, I might not be applying enough consistent heat to the actual joint on the board. How do you tell if the pad itself is getting hot enough, or is it too early to worry about the tip health? I just want the solder to flow without me having to blast it with excessive heat because I feel like I'm always trying to find the right balance between the two.

The heat timing thing is what trips people up. You can have a perfect tip and if you don't hit the joint and the pad at the exact same time, it's going to be poor. My old fixed temp irons worked just fine because you had to learn the rhythm. When I was doing those tiny surface mount boards, I learned you don't just sit the tip on the part. Do you find the heat lingers long enough for the solder to flow or is it just a quick touch?