After all, that debt servicing is overwhelmingly us paying money to ourselves. Why would that be an issue?
Are you seriously arguing that money transfer from one group of people to another group of people is not an issue? Seriously? Have you paid attention to politics of any country for any period of time in any of the history of the world? Transfer of money from one group to another is probably the biggest source of political strife that I can think of (behind religion). Ignoring foreign held debt, one could argue that interest payments are just shifting money from one US pocket to another US pocket. But, if you happen to be one of those US pockets where the money leaves (i.e. a taxpayer) then in many cases people do care and they do consider taxation, spending, wealth transfers to be a very significant issue.
National debt in of itself is not a problem.
Countries need debt for day-to-day operations, to finance large projects, and for buffers against significant disturbances (wars, famine, pandemics, etc). Imagine if we had a zero-debt policy and a country attacked us, would we just sit back and say, "well, we lose since we can't go into debt to fight back". You are essentially making a false dilemma fallacy. We have more choices than (a) zero debt and (b) our current path of increasing debt for the entire foreseeable future.
Unsustainable national debt is a problem though. You are partially correct that an aging population is a big portion of that problem. As the baby boomers go from being primarily tax payers to primarily tax recipients, this debt is going to go up significantly. But, if you read the GAO report that the OP's graph comes from, there is much more to the story. The unsustainable portion is actually the interest being paid. We could handle the aging population without much change to our taxes or spending. But the interest portion is currently unsustainable and is leading to that massive increase in debt/GDP predicted in the OP's graph. That will require changes to our taxes and/or spending to address.