I'm not a coach, nor do I study this stuff for a living, so I can only speak from my own experience. I'm a 32yr old ex distance runner who used to despise weight room training. As a collegiate athlete, I was VERY injury prone. Stress fracture after god d*mn stress fracture. I did everything I could to stay out of the weight room, even when my coach occasionally held weight room sessions.
As a 32 year old, I am no longer training for peak performance, I still run 4-10 miles on most days of the week, and I like to do a good hard effort road, xc course, or track workout once every 2 weeks. And only in the past year have I actually felt REALLY good on most runs, and the only thing I can chalk it up to is a regular weight lifting routine.... actually a lot of heavy lifting, 4-5 lifting sessions per week. It isn't like I have been training for a PR, so who knows, maybe lifting weights would be detrimental to getting a PR. All I can say is, I have felt like a million bucks, consistently, for 6-8 months, and have actually had some darn good workouts (relatively speaking). No injuries (but that could be due to less mileage and better diet), and its hard to explain but I just feel 'stronger' during runs (more below). Psychologically speaking, I feel fresh mentally, because I actually look forward to getting into the weight room. (which may or may not apply to a distance runner training for peak performance, as you need to spend a lot of your time running). As a physiologist, I am more than aware (if that's possible) that biological variation between humans is HUGE. And this includes adaptations to training stimuli. Other words, just like anything else I've seen in science on human subjects, weight lifting may help some runners, and not so much others. As for a mechanistic theory behind how tf weight lifting (aka, maybe 8-15 reps X 3 or 4 sets, plenty of recovery between sets, and usually maxed out by last rep of last set) could improve distance running performance?? I have NO CLUE. It doesn't sound logical to me.
I used to despise weight room training as a distance runner for two reasons. One, I was terrified that I was going to get bulky. I'm naturally already on the bulky side, relative to most distance runners, as I am 5'9" and my ideal running weight is 145-150lbs (I now weigh 165lbs). If you're the average distance runner however, you likely don't have to worry about that. Most distance runners cannot for the life of them gain muscle mass. And even if they do gain some, it MAY still be ok, depending... Two, I was worried about the exact same thing you are worried about - WASTING TIME. Mainly because I couldn't fathom a mechanistic explanation for how spending time in the weight room is any better than getting out and putting one foot in front of the other (....the law of specificity), or even resting between workouts. it seemed like a waste of time. Since I have been lifting, however, I feel much stronger during runs. I know I know, 'duh, you're lifting weights, of course you'll feel stronger.' But I mean I feel stronger during a 10 mile run, and in a good way. Not in a bodybuilder too heavy kind of way. Perhaps, 'fatigue resistant' is a better term. But how how in the hell could my lifting routine (heavy weights, plenty of rest between sets, only 8-10 reps per set) help with fatigue resistance during a 10 mile run??? Doesn't seem logical, but at the end of the day my lifting routine is clearly helping my running.
My advice is this - First, definitely figure out a 'core' routine and stick to it, do it at LEAST 3 times per week. If you don't know where to start, talk to people, email people, Google. Just anything to strengthen your core muscles and it doesn't need to be a lot. As for weight lifting... I don't think it will hurt to try. Start with 2-3 sessions per week. The 21 year old me would punch 32 year old me in the face for saying that. But if it doesn't help, I think you should be content in knowing that it was worth a try, especially if you're not happy with your current performances.... What have you got to lose? Don't allow time in the weigh room to take away from time putting in mileage or quality workouts, unless those aspects of training need to be reduced anyway, and definitely get enough rest/sleep, and definitely eat quality foods- Because lifting PLUS running will take it out of you. Lifting isn't for every distance runner, but I do think it is potentially significantly beneficial. And if you're still in school- Unless you're from a really poor family and you actually have to work a full time job to support yourself, then trust me, there is enough time to waste, and it will be worth a shot. Honestly- Worst case scenario, your running doesn't improve. But I don't believe that weight lifting will make it much worse.