I agree that we should be tracking years of lost life expectancy. I also believe in attempting a quality adjustment; we get better public policy from counting Quality Adjusted Life Years (QALY) lost.
I think that the main obstacle to this approach is that people instinctively think cross-sectionally rather than longitudinally. When you say "kills young people" and "kills old people", listeners focus on 2021. A young person is somebody born in 2000 and an old person is somebody-else, born in 1940. That is cross-sectional thinking, with 2021 as the date of the cross section.
But the real issue is that we nearly all live to grow old and get to experience both sides of the issue. Someone born in 2000 is likely to live until 2080. They will probably see a different respiratory virus going round then; it may even be what kills them.
So what would you personally prefer? The pro-lockdown approach trades losing a year out of your twenties to lock downs, in return for some action (perhaps ineffective, perhaps unnecessary) to ensure that you survive the plague of 2080 and get to live a few years longer.
The anti-lockdown approach makes the opposite trade-off. It is not really young versus old; it is now versus later.
I'm getting old and rather dreading later. I would prefer that society prioritize letting the young get on with their lives, education, dating, etc. and if I lose some poor quality care-home years, well, I've seen my grandparents and parents get old and frail, and I'd rather not dwell on the details of what I'm dreading.
If young people want to argue for the anti-lockdown trade-off, if they claim that in general young people shouldn't be locked down in the hope of protecting the elderly and they will accept their fate gracefully when they get old, well, I would think them wise beyond their years.