So, I'm not familiar with your training history. Based on this post, it reads like you took you first Judo class, and then have a bunch of thoughts about "completeness" and whether it's the fastest way to learn takedowns. From context clues I see you also train BJJ, so if I put two and two together, you're likely a BJJ practitioner who is looking to import takedowns into their BJJ game. These are my assumptions, so if these assumptions are wrong, then most of what I say is going to be irrelevant.
The "completeness" of a grappling style is a phrase that I find puzzling. Do you mean "has the most options", or "most applicable to real fighting/MMA"? BJJ's propaganda (for lack of a better phrase) is that it is the most "complete" grappling art, because it has fewer rules and more options. However the notion doesn't really make any sense, because some options are rewarded and some aren't. Currently the rules prioritize some strategies and tactics over others, and a resulting metagame is created (see: current state of gi BJJ, which is primarily a guard and guard passing metagame). Having certain options available (i.e., takedowns) but not incentivizing them (all takedowns are 2 points, but you can pull guard and receive no penalty, so there's no requirement to play this game) isn't the same as being "complete". As for MMA, there's evidence of success for both Judo and Wrestling, so that is not applicable either.
If complete means "flexible in the rule sets of other martial arts," I think it's still a flawed premise. It's like asking "How long is a piece of string?" Well, as long as it needs to be. A martial art will only be as complete as its rule set. It's funneling an overall idea of combat through certain filters (i.e., no striking, no leg attacks, no leg locks, etc.). Judo is perfectly complete for Judo competitions. Sambo is perfectly complete for Sambo competitions. However, you get specialization into the rule sets that is going to not be applicable outside of that rule set. Any person who has competed in a Jacketed Throwing Art will have a base level of proficiency in other arts, and may even be able to defeat novice to intermediate practitioners. However, high levels involve specialization, and that's where the nuances in rules and attire become important. For example, knowing the principles of throwing is a cross-art skill; however, setting up those principles with the competition attire is a specialization. I've seen videos of Judo guys getting beaten by Shuai Jiao guys because of the little vest Shuai Jiao guys wear. However Shuai Jiao guys will lose out in Judo tournaments because they get sleeve gripped into oblivion. Even though Shuai Jiao jackets are the same shape as a Mongolian Wrestling Jacket, they get tooled in Mongolian Wrestling because of the stiffness of the material. Sambo gripping is totally different from Judo because of the rule sets; Sambists would be likely be DQ'ed in Judo if they didn't adapt to the rules. However, any of these jacketed throwing arts can teach you how to manipulate balance via hand grips on clothing, leading into a throw or takedown. This skill applies regardless of discipline.
Not having grips, though, changes the game. It requires body positioning mechanics that either aren't practical in a jacketed throwing art (no judoka is going to let you get double unders in the gi, you'll be gripped), or just simply aren't practiced due to rule variances. Judoka who train specific to sport rules won't be as adept at countering singles as other wrestlers are, for instance (in fact some query whether they ever were, hence why leg grabs were banned; I digress). However these motions are integral to the application of moving the human body, or taking someone else down, in the absence of clothing grips. There are some areas of similarity, sure. But they are different games fundamentally. Trying to say Judo is better at teaching you takedowns than wrestling is silly. Judo is better at teaching you takedowns...when you're wearing a jacket.
It's also disingenuous to train an art for the purposes of importing it back to another art. MMA training was thought to gravitate towards an "X art for MMA" training regimen, where we figure out what parts of each art can be cherrypicked into a "best practices" skeleton. However, this top-down approach has never caught on. MMA continues to be a bottom-up, "train wrestling for wrestling's sake", "striking for striking's sake," camp methodology, and then the fighter experiments with their toolbox in MMA sparring/training. To extrapolate the lesson, learning an art in the context of its own ruleset is going to give you a better grasp of the art than trying to learn it in the context of another ruleset. So if you're going to learn Judo, just learn Judo. Don't learn Judo...for BJJ.
Likewise, you're confounding "getting better at Judo" with "building upper body strength." If you want to build upper body strength, pursue a strength and conditioning routine. Barring genetic freaks like Marcelo Garcia, normal humans don't try to train strength/conditioning while trying to improve their technical skill level.
Finally, as for Judo being the "fastest way to learn takedowns," I'm unaware of any specific timelines that support this claim. A person who has been training Judo for a year will be able to compete, and likely takedown an untrained person fairly easily. The same can be said for the various wrestling disciplines. Judo, when it was first created, was a standout martial art because it was created by someone who took the art of learning very seriously. It was revolutionary at the time. However, it's patently false to think that this approach only exists in Judo. That same academic approach to training exists in wrestling as well. The allure of the olympics is too great to ignore the act and art of coaching/teaching. Unless you've experienced it first hand, it's easy to think wrestling is a conditioning contest (show me a competitive fight sport where conditioning doesn't play a role; even full contact taichi push hands involves conditioning), but there is a very large amount of structured technical training that goes into place.
To summarize this whole mess, you should try to train as many arts as you can, and compete in as many rule sets as you can. If you're achieving good results in multiple rulesets, you can call yourself a good grappler. If you don't have the time/energy to do that, then just train whatever is fun or enjoyable to you, because it means you'll keep doing it. No matter how much you try to force it with a "I know this is the best art" mindset (which as discussed isn't true anyway), unless you like what you're doing on a fundamental level you just aren't going to stick with it.
