I like that some people are thinking outside the box. When a runner seems slower than they should be for the work they are doing, they just might be missing something that the average runner has going for them.
When I started running 10 years ago at the age of 28, I was 11 years removed from being very active in high school, and developed some deficiencies somewhere. I almost immediately could run something like a 21 minute 5k (after a month), but then I stagnated there for around 9 years, despite working up from 12 miles per week to 50 miles per week. My 5k never went under 20 minutes.
I did what everybody said...if you can't run such and such speed, you need to worry about building miles. You don't need speed work. So I ran miles, did the occasional tempo. Did maybe 5 random track workouts spread throughout the whole year, and I didn't get fast.
Last year, coming off of an odd injury, I decided to work low miles. 12 to 15 per week. Within those miles, I was running 8x200, 6x400, 5x800, and a tempo. All within a 2 week period. Within 3 months of that, I easily went under 20 minutes. I just needed to wake my body up and move like an athlete. Something I hadn't done very much of in almost 20 years. The power was gone and I needed to bring it back. Slow running will not bring that power back.
I don't think the weight room is necessary either, but if that motivates you, do it. If you loathe it, it will give you a reason to seek avoiding your workouts.