I think part of the explanation to your question is because your order of events is slightly backwards.

It's quite true that the Red Army (and the United States military!) participated in WWII and liberated concentration camps, and that the Red Army reached Auschwitz by virtue of geography. And relations with the Soviet Union were far more involved than that; while Russia was certainly under a different government, early Zionist leaders like Herzl had reached out to and received support from Russian leaders for their national movement in the early 1900s too. The early leaders of Israel were socialists too, and held some ideological overlap with the Soviet Union, leading to some initially warm relations. Then we start to get to crunch-time, and events played out somewhat like you suggest, but somewhat differently too.

After WWII ended, Soviet minister of foreign affairs Andrei Gromyko — otherwise infamous for his obstructionism — approved the 1947 UN partition plan for Palestine.

He certainly announced his support for it, that much is true. But the United States did so two days earlier than the Soviet Union, announcing their support for the majority proposal of the UN Special Committee on Palestine that pushed partition. However, it was not the Soviet Union who twisted arms and put diplomatic capital on the line to support the partition plan, to the same extent. The Soviet bloc certainly voted in favor of the plan, but the United States used threats of aid cutoffs and also allegedly bribes to convince some states to vote in favor of the deal, often in a way at odds with what the State Department recommended, because President Truman was good friends with Zionist leader Chaim Weizmann and felt a great deal of sympathy for the Zionist cause. This personal closeness didn't necessarily match on the Soviet side, but that wasn't determinative.

In the ensuing war, Czechoslovakia provided the Zionists with military equipment while the US didn't.

This too is true! The United States in fact attempted to implement an arms embargo on the region, which was viewed as a slap in the face to the Zionist leaders (though it may have helped them, in retrospect). This sore spot, however, paled in comparison to what was to come in the Soviet Union.

And when the state of Israel was finally established in 1948, the USSR was the first country to recognize it.

This is incorrect. It certainly provided de jure recognition to Israel first, about 3 days after Israel declared independence. However, the United States provided de facto recognition within 11...minutes. Both are different but functionally meant that Israel saw both as helpful and partner-capable.

Yet, Israel never aligned with the Soviet Union — quite the contrary. This enraged Stalin, who brutally turned not only on Israel but also on Jews in general whom he now suspected of disloyalty. By the '50s, Israel was so solidly anchored in the Western camp that it staged a joint invasion of Sinai with France and the UK in response to Nasser's nationalization of the Suez Canal. From then on, the Eastern Bloc would align with Arab states and endorse aggressive anti-Zionism until the bitter end.

This is where things get a bit funky in your telling.

The sticking points began long before the Israeli decision to turn away from the Soviet Union. There were indeed sticking points over the Israeli system of government (democratic, as opposed to the Soviet model). However, there were breakdowns in relations almost immediately, and indeed even while the war was ongoing, because the Soviet Union had already adopted some questionable policies.

By September 1948, while the war was still ongoing, the Soviet Union was warning Soviet Jews not to identify with Israel, evoking dual loyalty tropes. The Soviets were afraid that a national movement like Zionism might "infect" Jewish populations in the Soviet Union. Naturally, this was something the Israeli leaders noted and disliked; by contrast, they could travel to the US and raise over $100 million in donations from American Jews who were just as enthusiastic as many Soviet Jews about the establishment of a Jewish state. The Soviets' official press, by the end of 1948, was already running articles about how the "bourgeois elite" of Israel was awful, and Israeli attempts to cultivate ties with Soviet Jewish communities led to a rebuke of Israeli diplomats by the Soviet Union by early 1949.

Israel, during this period, was asking that the Soviets allow Jews to emigrate freely to Israel if they so desired, as the United States was willing to allow. The Soviets didn't just oppose this, they allegedly retaliated against Jewish communities that expressed desire for this, and the Soviets used lists of Jews who wished to emigrate provided by Israeli authorities to find targets. You have to keep in mind that nationalism as an idea was semi-compatible with the Soviets in the context of the Holocaust and the hope that Israel would adopt a more Soviet-style model, but it decided to become more democratic from the start and the Soviets feared "contagion" of these ideals (in part because of the dual loyalty tropes of antisemitism, of course). The Soviet decision itself to support the establishment of Israel was, in part, to force the British out of the Middle East and to try and sow the seeds of a division between the US and the British, which it actually succeeded in doing (which culminated eventually in the Suez crisis, and the Anglo-Egyptian negotiations in the 1950s, among other events). Once the British were out of the area, Israel had served its purpose.

Israel did not take major steps to align itself with either side at the start. But it was clear which was more willing to be friendly in 1949, after the war had ended. The Soviets and Israel would cooperate on some points; the Soviets supported Israel's admission to the UN, while Israel supported communist-leaning states seeking recognition in the UN themselves. But the Soviets ultimately, as one expert put it in a 1949 Soviet conference on "colonial and semi-colonial countries", should turn to the Arab countries where "all the objective conditions for a new upsurge of the national liberation struggle are currently prevalent." In short, the Arab states were far more valuable allies, and the Soviets saw that from the get-go. Thus, the Soviets adopted a neutral policy towards Israel that gradually cooled from 1949-53, and the Doctors' Plot and other antisemitic incidents in the Soviet Union helped contribute to that cooling from the Israeli side. But the Israeli side did not act against the Soviets to inspire this change; the Soviets decided that they'd done what they wanted with Israel and had bigger desires than an alliance with it, especially given the "threat" of Soviet Jewry becoming more strongly Zionist.

Sometimes people point to 1950, when Israel stopped being neutral diplomatically and supported intervention in Korea, as an explanation for the Soviet change of policy. But the Soviets were already shifting policies, as I said, and already adopting both antisemitic views in how they dealt with their Jewish population and also in how they dealt with Israel, while the US was becoming more friendly. The US itself took more time to warm to Israel, because it too wanted to target the Arab states for alliances, but it failed to succeed in playing for both. And the Soviet reaction in media doesn't bear out that Korea was the reason for any Soviet shift; instead, it was already coming, and Israel could do little to change it...nor did it mind, since it eventually found allies in France, and then later the United States, sufficient to give it the basic access to weapons needed to defeat Arab states in war and survive. After all, the Soviets had voted against Israeli interests at the UN on Jerusalem in December 1949, before the Korea vote was anywhere close to up.

Thus, it's important to not flip the script: it's important not to blame Israel and Jews (or their actions) for the antisemitism of Stalin. That antisemitism already existed long before Israel had shifted its policy, and was prevalent among Soviet leaders, despite their claims to the contrary. Stalin did not brutally turn on Jews because of Israel's shift in policy. He was already suppressing Jews because he feared "disloyalty", itself based on antisemitic tropes that if Jews supported Israel's establishment or existence that they'd subsequently become less loyal to the communist cause and betray it for the "bourgeois" Israeli leadership.

So Israel didn't ignore and turn down Soviet courtship: the Soviets turned down Israeli desires to remain neutral, and decided not to court them to break that neutrality in the Soviet direction. As they courted the Arabs more, Israel naturally gravitated more to the West, which was cemented later on.