Your common sense is reasonable enough, but you know that some things are counterintuitive.
Yes, you should warm up. Yes, it improves performance for nearly all runners, especially runners who are reasonably well trained for their event.
If you're at all trained for 5k, you're not going to wear yourself down with a modest warmup. Add to that the potential benefit of what you can do with running economy/oxygen uptake: pros outweigh cons.
If you look online and check out a few good sources for good 5k warmups (or I expect some coaches might reply here to offer ones), you'll find some that might work for you. If you're skeptical, start with the one that looks the least taxing and see how that works. "Look online" applies generally. The search tool for these boards should steer you in a good direction too.
Yes you should warm-up. Consider this - it is a warm up and not a wear out. If it tires you out then either you are not doing it right or you are not yet in shape.
There are ways to test your warm-up. One way is to look at your repeat workouts. Often the first couple are the slowest. That is because you are not fully warmed up yet. Another simple test is to go into the weight room with no warm-up. See what the maximum you can bench press once. Then go out and jog a mile. This should be a jog meaning at least 2 minutes slower than your mile race pace. Then do the bench press test again. I can almost guarantee that you can bench press more after the mile jog. If you don't need a warmup and if the warmup tires you out then you shouldn't be able to bench press as much after the mile jog.
I assume you're trolling, but if not have you ever ran began a run at significantly faster than your easy pace? So for a runner who typically runs 7:20s you'd run somewhere between 6:30 and 6:40, and you feel like absolute dogshít after about half a mile in, and that feeling never goes away and your heart rate stays high, while if you settle into it after a couple of miles you're cruising along.
So, you don't want to go into your race feeling like that
Warming up makes a huge difference for me. It helps you to get in the zone to race mentally and physically prepares your muscles so you are ready to race.
If you are tired after your warm up mile you are running it too fast. We run our warm up at 9 min/mile pace, much slower than any normal runs even at easy pace. Doing it right before your race could also contribute to feeling like it makes you tired. If you jog 45 minutes before then do some drills and stretches you shouldn't feel tired before the race.
Warming up makes a huge difference for me. It helps you to get in the zone to race mentally and physically prepares your muscles so you are ready to race.
If you are tired after your warm up mile you are running it too fast. We run our warm up at 9 min/mile pace, much slower than any normal runs even at easy pace. Doing it right before your race could also contribute to feeling like it makes you tired. If you jog 45 minutes before then do some drills and stretches you shouldn't feel tired before the race.
What are your PRs? They're probably quicker than mine so by that logic I would need to go even slower