It is not because of the cultural revolution. That was from 1966 to 1976. There was the 1964 Scheme of Simplified Chinese Characters, but that was just a final step of officialness. More on that in a minute. The idea of simplified characters as the norm first started gaining speed in the 1950s. However, both in mainland China and in Taiwan, the simplified forms already existed. The following is adapted from answers I've given on this before.

It's important to note that the simplifications weren't a CCP invention, and the vast majority of those which were eventually made official were already widely used, some even coming from simplifications in non-Mandarin languages. But we need to go even earlier.

Reforming the writing system, as well as the standard speech, was something that has been pushed for well before the founding of the PRC. They just continued an effort at simplification that had been underway for some time, and actually that was less an effort at simplification than a codification of simplifications that people were already using.

For example in 1935, in pre-communist Republican China, the Ministry of Education introduced a simplification scheme of 324 characters. Other private efforts were also ongoing throughout the decade. And it's also not the case that they simplified the most common characters. The choices had more to do with characters that were in frequent use and in many cases ones that already had widely used vernacular simplifications, many of which are also in use in Taiwan which never saw any of the PRC reforms and where traditional characters are still the norm.

The majority of simplifications are taken either from calligraphic or hand-written forms of how the traditional form is (訁to 讠/ 東 to 东) or from "vulgar" characters, i.e. simplification that people were already using unofficially. A good example of this is 雙 shuang which means "two of" which is graphically related to 隻 zhi which means "one of". Instead of writing 隹 twice it's easier to just write 又 twice so people wrote 双 for shuang and now they still do.

That is to say, simplified characters in Mainland China both predate the PRC, and are also used frequently in handwritten forms in Taiwan. The standardisation of simplification, started by the Nationalists before 1949 and continued by the Communists (and pushed for by some among the late Imperialists) was really just saying "now this is official, you can teach it in schools". And if you go to Taiwan where people are otherwise generally using traditional characters, you'll absolutely see 双 written all over the place, such as on hand-written signs for places selling shoes.

Not only that, but before the KMT (Republican China / the Nationalist government) fled to Taiwan, they were also trying to make this all official. A full ten years before initially moving into Taiwan, in fact. 1935 saw the first formal effort by the KMT to simplify. This was a list of 324 characters, supported by a couple hundred scholars. Unfortunately for those behind the task, this was short lived and withdrawn after 6 months. There were a number of reasons for this. A few years prior in 1932 the Ministry of Education changed the standard spoken language to more or less what it is today, being a slightly more literary version of the way educated people from Urban Beijing spoke (not actually Beijing dialect as people so often say).

The decade was intended to be a pretty significant time for language reform. However 1932 also saw the first hostilities (the January 28 Incident) of what were to become the Second Sino-Japanese War and then WWII in Asia, and the renewed tensions with Japan, in addition to Communist insurrections in the South and growing elsewhere, the KMT government had to abandon a lot of their plans relating to language policy.

So, it was shelved. However in 1951, two years after the CCP took control, simplification was once again called for. Actually, this call by Mao to simplify writing was not a push to simplify characters. Rather, he wanted to see a home grown alphabet be put into use. Mao's dreams of a Chinese alphabet didn't ever materialise, though the previous (and in Taiwan continued) use of the Zhuyin script (注音符号) could be argued to fit that bill.

Four years later (1956) the CCP government began the so-called First Scheme of Simplified Chinese Characters. This got updates over the years, but can be generalised as Simplified Characters as we know and tolerate/love/detest them today.

It wasn't until 1964 that the official simplified character set was published. It covered 2,236 characters, many derived from the ways I mentioned above.

The really interesting thing happened in the 1970s though. In the middle of the decade (1975) the Second Round Simplified Characters were submitted by the Script Reform Committee for approval by State Council. This was 248 new characters to be used immediately and 605 proposed characters for eventual adoption. Then 61 of those were also components of approximately 4500 other characters which would then be modified as a result. This was the most ambitious push for simplification yet. Two years later in 1977 the list was published, and then just 9 years later it was retracted. The reason it's so interesting (to me) is that you still see them in use. People who were educated during this time have a good chance of having learned them, so it's not at all impossible to see people writing 仃 for 停 or 歺 for 餐, so long as they're in the right age group.

There are two major reasons this was retracted. The first is that it was taught inconsistently. Like earlier efforts at standardising pronunciation, teachers weren't equipped or trained well enough. The bigger issue though, and the one that gets us back to your question is this:

Unlike the first round of simplifications, these were characters no one already knew. See in 1964 the government really was just saying "hey you know how you're already writing 让 for 讓, even though that's not even based on Mandarin? Well you can start doing that on your homework and in official documents. You're welcome."

Simplified Characters weren't new. They were just being given official status. Prior to this it was more like what we see in Japan with kyūjitai 舊字體 / shinjitai 新字体 character forms, where yeah now it's the standard but the adoption of these forms was so gradual there's not really a clear break that people point to as "yep this is the day we all started writing differently". They were a slow change over generations. With simplified characters, it's the same idea, except that in 1964 it was made a formal thing and added to the bureaucratic machine.

The way that the government managed to get everyone to use them is that it didn't have to do anything to make it happen. It just had to say "yeah ok you can do that" and then got the schools to teach them in a consistent way.

The reason Taiwan uses one system and China another has less to do with reforms in China, and more to do with the fact that the Nationalists got caught up dealing with bigger more pressing problems in the 1930s and never managed to get their own reforms pushed through, while in the 1950s the Communists were able to pick those same reforms up and run with them while at that point, Taiwan was again busy with other things, both linguistic and otherwise.