In the usual case, then, you have nine pitchers contributing variously to five categories: wins, strikeouts, ERA, WHIP and saves. Starters will do better in wins and strikeouts, while closers get you saves. Closers are typically better than your average starter in ERA and WHIP, but have less of an impact on the category due to their low innings counts than your top starters. While it's important to get as many total points as you can in all five categories, it's not important how or when you get those points.
For example, if you have nine starters and zero closers, you'll likely find yourself ahead in wins and strikeouts and behind in saves. If you have seven starters and two closers, you'll likely be competitive in all categories, but derive no particular advantage in any, and if you have six starters and three closers, you'll be strong in saves, but it'll be harder to compete in wins and strikeouts. To that end, if you miss out on an early closer run in the draft, there's no reason to reach. Just fill up with quality starters instead and build substantial leads in wins and strikeouts.
That the league has become more pitcher-friendly in recent years makes this strategy even more viable, as it's easier to find No. 8 and No. 9 starters who get you wins and Ks without hurting your ERA and WHIP.
Nonetheless, while I'd advocate punting saves where they're too expensive at your draft, that doesn't mean you should tank the category altogether. If you've rolled with nine starters, you should have big leads in wins and Ks in June or July, and you can trade some of that surplus for closers. In fact, there's nothing wrong with having no closers for the season's first four months and five closers for its last two. It doesn't matter when you get your stats, only that you get them.
And it's typically very easy to trade for saves as closers help you in that one category only. So if a Team A is 10 saves ahead of the next team down, and 15 saves behind the next team up, its owner will gladly move his best closer for a quality starting pitcher as it costs him very little and might benefit him quite a bit. This is rarely the case when you're trying to trade for power, as an owner who is fairly set in his slot for home runs will also suffer in RBI and runs by trading a slugger. The same can be said for acquiring top quality starting pitching, too - someone could be set in wins, but trading his ace will cost him a lot in strikeouts or ERA. But trading for a closer is far easier.
In sum, there's no reason not to draft reasonably priced closers who fall to you in an appropriate slot. But there's never a reason to panic or reach should you find yourself stuck on one side of your draft and witnessing all the quality closers flying off the board.