What you're failing to understand is that this year would have been more like '96/'97. I don't know how old you are or if you were even around back then, but we all knew the next year that the Spurs would be good again.
Not making the playoffs this year doesn't mean were going to miss them 2-10 years in a row. As long as Kawhi is still on the team and comes back healthy next year, we'll be right back to being an outside le contender again, but with the added bonus of having a much better draft pick to add to that.
The other example I always think back to, that really pissed me off a few years after it happened, is when the Lakers missed the playoffs in one of Kobe's prime seasons and got Andrew Bynum out of it (who ended up being a key piece that helped them win two more Championships).
Who gives a about the streak. TD isn't here anymore. All those records and streaks were his. If TD was still here, I'd want the streak to continue because I want people to see how great he was and all the hard work he put in, but now that he's gone, I don't care about the streak much at all. I look at it like the Lakers with a prime Kobe after Shaq left. The Spurs need more talent if they're going to be a serious le contender again, and the best way to get that is getting a higher pick than they would normally, possibly even a lottery pick as that would go a long way towards helping them get back to true le contention.