V̇O₂max work rate can typically be held only for around 8 minutes or so, so you're not going to be racing at V̇O₂max pace. And while the pace will be "close", it's in the nature of endurance that small reductions in work rate lead to much longer times to failure.
Advanced 5k training programs often recommend high mileage. For example, 20km a week.
20km a week is not high mileage. Assuming an average training pace of 5:20 min/km, that's less than two hours of training a week.
Would focusing mostly on high intensity workouts be as effective?
For races of around 20-25 minutes, I am a fan of Seiler intervals (4 x 8 minutes) - see Seiler's paper from 2013 or this use of it for cycling training.
Would focusing mostly on high intensity workouts be as effective?
From your question, I'm assuming you are not advanced/elite, which is where nuances may occur to make gains over rivals.
The 5K training programs out there don't really vary that much in what they suggest for beginners and intermediates. While I can't point you at papers about this, those programs have been promulgated for decades, and have shown to be effective for many thousands of people.
If, in general, focusing on VO2 max workouts was best, that's what we'd be doing.
Without an endurance base, it does seem that - while it's possible to reach a reasonably high level of performance - such gains can't be maintained over a period of time. Look at Chris Carmichael's Time-Crunched Cyclist or Matt Fitzgerald's 80/20 Training - an endurance base does seem to be necessary for consistent, long-term performance in endurance sports.
(Note that the fact that just doing HIIT can keep you as healthy as long steady cardio and even help with weight loss does not contradict this - here we are talking about an athletic performance, not being healthier than the typical sedentary western adult.)