Ignoring power is a bad idea, but assuming for the sake of argument that we do, I'd still put Hank Aaron above Pete Rose. Rose is the all time hit king, but Aaron is third on the list, which isn't too shabby, and his career batting average is .305 to Rose's .303. Pete Rose has more hits than anyone else, but he also made more outs than anyone else (in fact, 1,192 more outs than anyone else).
As "pure hitters" go, I'd still rank them below Ty Cobb, who I'd rank below both Babe Ruth and Ted Williams. Cobb is second on the all time hit list, and he has the highest career batting average at .367. Neither Ted Williams nor Babe Ruth are even in the 3,000 hits club (though Williams would have been had he not fought in two wars). Ted Williams is 7th all time in batting average (.344) and Babe Ruth is 9th (.342). So you'd think that I'd rank Cobb above Ruth and Williams, but I don't. Here's why.
"Pure hitting", in my definition, is the ability to not get put out. If you go to the plate, and you're put out, you failed. If you reach base, you succeed. Personally I think home runs are better than singles, but we've eliminated that from the discussion, so I'm forced to improvise. I also think singles are better than walks, because singles result in RBIs more often than walks, but again this isn't part of the discussion. So getting on base, regardless of how, is equal as far as hitting goes.
Ted Williams' career OBP is .482. Babe Ruth is .474. Ty Cobb's is .433. Williams and Ruth are 3-4 on the all time walks list. Ty Cobb is 49th. The guy did not talk walks.
As far as Ichiro goes, his career average is .333, which is nothing to sneeze at, and I think he's a first ballot hall of famer, but it's lower than that of recent players like Tony Gwynn and Albert Pujols, and Ichiro doesn't walk either (188th on the career OBP list).
To address some of the other choices submitted, Joe DiMaggio had an incredible 56-game hit streak in 1941, but his career average is lower than Todd Helton, and his career on-base is lower than Brian Giles. Derek Jeter's career batting average is lower than Matt Holliday, and his career on-base percentage is lower than Dave Magadan. Wagner ranks below Helton in BA and J.D. Drew in OBP, so he's out of the discussion as well, but Rogers Hornsby has a right to be included in the conversation (2nd all-time in batting average, 8th in OBP).