No, alphabet based systems aren’t easier to write. It simply shifts the difficulty from one aspect of writing to another.
Let’s take English for example. There are 26 Latin letters used in the language, and words are formed by placing one letter next to another, from left to right. But it is extremely difficult to spell English words. You have to remember the sequence of letters that form each word.
In Chinese, each character needs to be remembered. But once you’ve remembered the character, that’s it. You’re done. A system of radicals exist to help you remember how to write the characters, but the radicals are secondary. The main unit is the character.
That is, logographs require you to memorize lots of different characters. Meanwhile, alphabets require you to memorize lots of different sequences. It’s the same difficulty, just placed differently. This is exemplified by the same literacy rate in China and Japan as the best alphabet-based nations.
The only reason Chinese seems difficult is because most places use alphabets. Don’t underestimate just how much 12 years of mandatory schooling has conditioned you to think in a very specific way that is tailored to reading and writing English in a very fast pace. If instead you yourself used a logographic writing system, learning Chinese would be easier than learning English.
As for number of words, it is no longer important. The library of all words, whether English or Chinese, is far too large for any individual to ever remember anything more than a mere fraction of them. Most coining of new concepts is done through descriptions anyways, and reduced via literary tools if they are popular enough.