Are distance runs more important than speed workouts when training for a 5k run?

I'm not a coach or much of a runner, but in the Army we ran 3 times a week. Two distance runs and a speed run. The distance runs were 3-4 miles at ~10 min/ mile and speed days were 60/120s or fastest mile. If a coach or runner says suggests something different than me, listen to them.

I'm not a coach either, but as a general proposition, in order to run fast, you have to train fast.  However, you can't overdue your speed training--it's one of the quickest ways to get injured (no pun intended).  So you should do no more than one hard speed workout a week, and focus the rest of the time on getting the distance in you need to run a 5K.    That's not really that much distance, compared to training for a marathon or a half.  You probably don't really need to do any distance runs longer than 5 miles when training for a 5K--that's already a third of the distance longer than the 5K.  Furthermore, at least once a week, your runs should be relaxed enough so that you don't feel like you're pushing yourself too hard, because you need to rest.   I'll defer to any coaches out there, but this is how I approach it.

Distance runs are, hands down. They are the bread and butter of all distance training from 5k to marathon. Basically every time you go on a long run it increases the ability of your lungs to harvest usable oxygen from the air we breathe as well as increase your muscles capacity to withdraw this oxygen from your blood. The oxygen, as you probably already know, if used to clean the lactic acid from your muscles that is produced while running. On the other hand speed work works on your tolerance of lactic acid. The more distance work you do early on, the faster you will be able to run later.

Both. You need to keep a “distance” run in your workout to keep the different physiological system in place that is required to run a 5K. I used to to do 2/3 hard workouts (reps, intervals, thresholds) with 4/5 distance/recovery runs per week. That’s quite a bit of training though and I’ve substituted most of my recovery runs with cycling and swimming these days but still maintain at least 1 or 2 distance runs in the schedule.