I don't disagree that core specific training is unlikely to help one's runner unless perhaps they are a high hurdler but core strength helps in many other ways as part of one's functional strength. As one ages it helps with balance, it is absolutely essential as a rock climber and it has certainly helped me keep up with my toddler and avoid back pain when I used to have rock him to sleep or he wanted to be held for hours.
I think weight training is different however. Lifting weights might have a marginal impact on one's marathon time but for the 5K and shorter distances I have found it has had a significant positive impact on my speed especially for the 400 meters. I think sometimes people associate weight lifting only with bulk when in fact weight lifting can make one stronger without a huge increase in body mass. If you just want to get ripped at the gym and have bulging muscles that isn't likely to make you any faster but if you want to target specific muscle groups you can absolutely increase your speed and power.
I can bench 240 lbs but my upper body while decently defined is not ripped and even wearing a t-shirt you wouldn't guess I could bench 150% of my body weight. That's because I focus on shorter lower rep workouts at higher weight 3 days a week vs. do long high rep sessions to build muscle mass. Now my bench doesn't help my running but my squats and deadlifts certainly have. My thighs are more like runners thighs than weight lifters or football players but the power and explosive speed is greater since I started lifting. Maybe this is just a post 40 thing and had I lifted when I was younger it wouldn't have made much of an impact.
That said I am not an elite athlete or even a super talented runner but I do wish I had lifted when I ran the 400 and mile in high school as I think it could have improved my time by a few seconds and that would have made the difference between being a runner who racked up a lot of second and 3rd place finishes at dual meets to one who actually won a fair share of his races.