The thing that you might not be aware of is that Ozzy Osbourne has zero musical ability. None. When he was a member of Black Sabbath, he was given writer's credit on all of the songs because they had decided as a band that each member would be credited when it came to songwriting, even though Iommi wrote all of the music and Geezer wrote most of the lyrics. Ozzy didn't lend anything except his voice to the finished product and his input never went much beyond some minor suggestions in terms of arrangements when it came to the composition process.
When Ozzy went solo, he needed a guitarist who could take his minor suggestions and coax actual songs out of them. Randy Rhoads wasn't a nobody like many people like to claim that he was. He had already made a name for himself on the L.A. club circuit with Quiet Riot before Ozzy decided to audition him. In fact, Randy was being compared to Eddie Van Halen who was the lead guitar player for a band that had opened for Black Sabbath on the "Never Say Die" tour. So Ozzy knew who Randy was and was well aware of what he could do.
"Blizzard of Ozz" and "Diary of a Madman" were both a huge success for Ozzy because Randy was such an original player. He had been a child prodigy who began playing at the age of 6, could play multiple instruments and had a very distinctive sound and approach. His classical training enabled him to blend that style with hard rock and heavy metal and the end result was something completely revolutionary. Ozzy didn't sit down with Randy and help him compose legendary songs like "Crazy Train" and "Flying High Again." Randy wrote all of the music by himself and Ozzy contributed his standard minimal input. The rest of the band helped Randy work out arrangements and Ozzy wrote his own lyrics.
Even if Randy had never died in an accident, it's likely that he wouldn't have been playing with Ozzy for much longer. He had succeeded in what he set out to do. He had managed to blend the classical influences that he loved with the hard rock and heavy metal that he had made a career playing. As a musician, he wanted to learn more, to progress further, to take his personal development to a higher level, and that meant going back to school to study music. It's likely that Randy would have abandoned hard rock and metal and steered himself in a completely different direction. Maybe he might have cut one more record with Ozzy. He wasn't keen on the way Ozzy treated his musicians. At all.
Originally the BAND was supposed to be called "Blizzard of Ozz", and the members figured that the fans would naturally shorten their name to "Blizzard" the way Black Sabbath fans referred to them as "Sabbath." But Ozzy and the record label decided to call the band "Ozzy Osbourne", and that was the name that was emblazoned across the sleeve when the record was released. Then, over the years, Ozzy would find reasons to screw his bassists and drummers out of royalties and songwriting credits, he would even go so far as to have their parts re-recorded. A lot of that had to do with Sharon being a Cvnt with a capital "C", but Ozzy went along with it.
Jake E. Lee was never asked to play like Randy. He was hired because, like Randy, he was capable of writing songs in the style Ozzy wanted to continue with, and he was good enough to play the songs that Ozzy already had to his name. Jake E. Lee could play Randy's songs and solos note for note just like Zakk Wylde could play Jake E. Lee's songs and solos note for note as well as Randy's.
Look at how Ozzy allowed Zakk Wylde free hands to blend all of that Southern inspired shíte-kicking nonsense into the songs on "No More Tears." Ozzy was content to allow his guitarists to play whatever they thought sounded good so long as it didn't stray too far from his main themes.
To hire a guitarist who's already well known, well respected, and well established wouldn't suit Ozzy's purposes at all. Iommi's ego was one of the main reasons Ozzy was fired from Black Sabbath. There were a few different reasons, but that was one of them. Jimmy Page and Ritchie Blackmore had already made a name for themselves. Neither of them would have been willing to become Ozzy's puppet. They wouldn't have agreed to write songs and give Ozzy credit for them. And they wouldn't have been willing to play on a bill under the name "Ozzy Osbourne" where they wouldn't receive any recognition. It would never have happened. Ever.
Ozzy has always sought out people who would appreciate being given a big break by being offered a position in his band. People like Jimmy Page and Ritchie Blackmore don't need somebody else to take them under their wing. Those guys are already famous in their own right, and rightly so.
I mean, look at it this way: A few years ago Zakk Wylde was jerking Ozzy around, using playing with Guns 'n Roses as a bargaining chip to get more money out of Ozzy. When people get established, they no longer allow themselves to be treated like rubbish.
Ozzy debated working with Buckethead, but things didn't pan out. After that Ozzy was convinced that he had the right idea bringing on people who hadn't got enough money or success under their belts to be arrogant yet.
No way he would have ever convinced Jimmy Page or Ritchie Blackmore to play for him, and no way it ever would have worked out if he had.