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This is just not true in general. The implications were very dependent on where you lived.

You should look up Paolo Sarpi who was a leading figure in the Venetian Republic who Galileo corresponded with frequently. Pope Paul V tried to have him assassinated at one point. If anything he became a more influential leader in Venice after the assassination attempt.

The freedom to work without the more extreme elements of the church getting in the way that was possible in the Republic of Venice at that time is one of the many reasons the position that Galileo opposed the church is hard to defend. If he opposed the church then why did he return to Florence in 1611 instead of staying in the republic.

If we were considering weather for example we could say that the weather is predictable over a period of 1 second, somewhat predictable over a period of 1 hour, and unpredictable over a period of 1 year. It doesn't matter how much historical weather data you collect you just can't say anything about the weather over long periods of time.

If a math writer says "the primes are unpredictable" this means something like if you were given a list of all the primes between 1 and some number N then there is no way to find the next prime after N that is faster than testing the primality of N+1,N+2,... until the next one is found.

The speed of the primality test doesn't matter for predictability. It doesn't matter if I can check the current weather in 1 second or 1 millionth of a second, it doesn't help me know the weather a year from now.

But not being able to work out the next prime quickly isn't the whole story. There are many things that we can say about the whole set of primes. We do know that the next prime after N won't be much bigger than N in some sense (see Bertrand's postulate). We also know roughly how many primes there are less than a given number (Prime number theorem).

There are also many properties that exist in small sets of primes (twin prime conjecture) and the OP's visualization shows some of these. So, much like the weather, the primes are not random while still being unpredictable.