I asked the same question Ismail, here is his answer.
Part of the purpose of his speech was to boost himself and discredit his rivals (Molotov, Kaganovich and other "Stalinists.")
The other part was to try to come to terms with the Stalin period in a way that didn't seriously rock the boat. After Stalin died and Malenkov took over, the Soviet press greatly curtailed praise of him, and the Short Course history he edited was withdrawn from publication but with no other party history taking its place.
So from 1953-56 there was a rather awkward situation where Soviet officials would continue to glorify Marx, Engels, Lenin and Stalin as equals but clearly treat Stalin in a somewhat hushed fashion compared to the first three.
Khrushchev's solution was basically shock therapy as carried out by a ten year old. Khrushchev's colleagues regretted the speech as soon as it began to informally circulate. Instead, the official summary of Stalin's misdeeds from 1956 onward was this text compiled mainly by Suslov: archive.org
The Soviet narrative of Stalin from 1956 until Gorby can be summarized like so: "Stalin was a Marxist-Leninist theoretician and revolutionary who defended the Party against Trotskyism and Bukharinism in the 1920s. However, got an inflated ego when the successes of socialist construction were attributed to him rather than the Party. He began consolidating power. In the 1930s there were widespread violations of socialist legality and groundless repressions which Stalin shared blame in causing. A giant personality cult developed around Stalin which distorted history-writing and created an unhealthy political climate. Then in 1956 the 20th CPSU Congress fixed this."
Khrushchev's speech wasn't published in the USSR until the Gorbachev period. The Soviets ignored it and referred everyone to the aforementioned Suslov-crafted text.
De-Stalinization absolutely could have been carried out in a more competent and honest way. But it wasn't, and anti-communists have made use of that ever since. Rather than an objective analysis of Stalin, Khrushchev presented his listeners with a fairly lurid exposé that, as I said, contained omissions, distortions and falsities
