"What’s your opinion on China’s logographic writing system in general (both simplified and traditional)?"
I can't stand simplified form. Traditional form is the form standardized during the Qin dynasty, thus should be recognized as a defining characteristic of China. The Sinosphere should be given special credit for being the last civilization to keep using the system that that was once (back in Neolithic times) widely considered the most sensible system:

We should view the switch to pronounciation-dependent systems as a form of post-Neolithic degeneracy.
"I don’t think that they should go with Pinyin"
I agree. The practical advantage of the logographic system is that the character is independent of how it is pronounced. Even someone born deaf can read logographic text without diminished communicative quality. Whereas with pronounciation-dependent systems:
Deaf learners face considerable obstacles to reading, but they also bring a unique skill set to the task.
Educators and psychologists alike have long debated the reasons why the vast majority of deaf children struggle with reading.
...
When hearing individuals read, they decode written symbols into speech sounds to recreate a spoken text. Learners read out loud, but even proficient readers create a “voice in their heads.” Accessing the meaning of a written word, then, is a two-step process: first convert the written item into spoken format, and then access the meaning of that spoken word.
This is not the case with logographic writing which does not require conversion of the written item into a spoken format in the first place, which is why logographic systems are superior: meaning is accessed in a single step.
"Maybe they could use a modified version of the Manichaean script."
"Or a modified version of one of the Indic scripts"
This would have the same problem as Pinyin in tying the character to one particular pronounciation, which is what we are trying to avoid.
"to create more positive relations with India"
The superior Indus Valley civilization also used a logographic system:

It was the inferior Vedics who put an end to it.
"I think it’s kind of inefficient to use a writing system that students in China are still learning as they basically go through the entire schooling system, instead of having it be a one and done affair."
The problem is with Westernized "New China" trying to force everyone learn it all, and at a fixed pace, which was never the original intention. In ancient China (where compulsory schooling never existed, of course), different sectors of society would learn as much or as little of it as they variously needed for practical purposes in their respective walks of life, and do so at their own pace. This is what we need to get back to.