The characters you're describing are commonly known as radical-phonetic characters, where part of the character indicates the sound, and the other part implies the meaning or category. It is my understanding that Communist China's push towards character simplification has broken a great deal of phonetic relationships inherent in this class of characters, making the language much more difficult to learn than it used to be, even if it is marginally faster to write.
Additionally, there remain the 10% of characters which don't fit into this mold, a great many of which are very common.
I don't know if simplification has broken a great deal of the relationships, but it certainly has broken some.
Here is a specific example:
Traditional Chinese
車禍 [che1 huo4] (car accident)
不過 [bu2 guo4] (however, but)
Simplified Chinese
车祸 (same meaning/pronunciation)
不过 (same meaning/pronunciation)
Notice the 咼 in the Traditional 過 and 禍. This phonetic component gives you some indication that is is pronounced like "luo, huo, guo, wo".
In the Simplified, you lose that relation, because you have the 寸 and 呙 units, respectively.
The phonetic components of Chinese characters don't always give you an exact reading, but they can help you get a good idea of what a character should sound like. There are exceptions, of course.
But even in that case the relationships weren't completely broken. While 过 changed, 娲wa, 祸 huo，涡 wo，窝wo，锅 guo，蜗 wo，etc. still share the radical to the right, and 过 is a very common character, you shouldn't need to guess how to read it.
I found that simplified characters are much easier to learn than traditional, it is just that much simpler. Enough of the phonetic relations are still there (and some new phonetic relationships were created, I think), and memorizing the base characters is much easier. On top of that, memorizing the characters still requires a lot of practice writing them, and simplified saves enough time that its definitely worth it. For example, for the character for far: 远（yuan), its traditional is much more complicated: 遠. On top of that, a sound relationship is still there, and it is much simpler.
I studied three and a half years of traditional characters, switched to simplified when I went to China, and then started studying Japanese, which uses a mix. I definitely am glad I studied traditional characters, but I feel at least for me, they are much much harder to learn, but that could be different for different people.
Out of curiosity, have you tried learning both simplified and traditional?
Only traditional. My knowledge of simplified, and the debate in Chinese academia about their real value, actually stems from discussion with Chinese linguistic experts, but I have no personal experience with simplified Chinese myself, except casually.
Sort of.
The characters you're describing are commonly known as radical-phonetic characters, where part of the character indicates the sound, and the other part implies the meaning or category. It is my understanding that Communist China's push towards character simplification has broken a great deal of phonetic relationships inherent in this class of characters, making the language much more difficult to learn than it used to be, even if it is marginally faster to write.
Additionally, there remain the 10% of characters which don't fit into this mold, a great many of which are very common.