Without the US, what benefit does the tpp provide member nations? Seriously asking.
This is actually a very good question. I will try to answer it to the best of my ability. I've been studying this subject for well over a year now.
Short answer:
Even though the U.S. is the largest market and the biggest player in said trade agreement, the size and value of the other nations is still a very "enticing" one. Plus, it also has to do with politics. You see, ONE of the many other core reasons that lead to the creation of the TPP is to counter China's ever increasing influence in the whole Americas region. Whether you care about it or not, the fact of the matter is that China has been meddling in what has historically being considered "America's backyard" (Latin America and Canada). And that is something that does NOT bode well for American politics in the region, its influence, its economy, and its people. China is a country with an entirely different set of values and objectives that do not align with what the U.S. wants, nor with what is best for the U.S. Imagine America being surrounded by socialist/communist/totalitarian states in the very same region. It's problematic, costly, dangerous, and overall stupid for the U.S. to let China's sphere of influence expand. And no, this isn't one of those cases in which "Nuke the motherfuckers" will cut it. Why?
Long answer:
OK, so, in economics and politics there's this thing called "Middle Power" country. This is basically a country who's economic and political power are not as powerful as a major world power, but it is still considerable enough to influence other smaller countries. Excluding Japan, the rest of the TPP countries fall into this category. Now, I know you're gonna say "Hey, wait a second. Canada, New Zealand, and Australia are first world countries/developed nations". And you're right. HOWEVER, the political and economical influence they have over other countries is limited or small. See, there's a huge difference between a developed country, and a world power. Living in a rich country is not the same as living in a powerful country. For example, Finland. Finland is a country with an amazing quality of life, BUT, its economy is small, and has no influence over other countries. Therefore, Finland is a developed, free nation, but not a powerful one. Now, there's more to it than just size when it comes to defining world powers. I encourage you to research this topic by yourself. I have provided you with a simplification of the term, so don't go taking this definition literally. Suffice to say that this is not one of those "Theories" some old fart wrote a book about to boost their academic prestige. Middle powers are an actual real thing, and do exist in the real world; these are not just unnecessarily complex terms.
Anyways, going back to the topic at hand: These middle power countries, their aim, aside from economics, is to form a "buffer" zone that limits China's expanding influence. You have to understand, even though the U.S. is the leading country for these smaller countries, these countries have ideologies and goals of their own; they do not need the guiding hand of the U.S. to tell them where to go and what to do at all times. The TPP and its push for approval, even though the America backpedaled, is an example of this.
So, for the rest of the world, the message is clear: "The U.S. has backed down. They are pulling in. Right now, America's priority is domestic, not international. What now? , What do we do now?". And, basically, the dialogue among these countries has been "Well, why not use what was left behind? Use it for ourselves". And, there is something people fail to take into account when thinking about other countries, and specially about middle power countries; and that is that they LOVE trade. In fact, they love trade so much, that their economies are oriented towards exports. This stands in deep contrast with the U.S. , since the engine of American economy is consumerism. This is why the U.S. leaving the TPP was seen as the final nail in the coffin for the TPP. America is the largest consumer market in the world. America was supposed to become the main engine of growth for the entire block. Basically, America would lead the TPP block and buy the most out of any country, and, in exchange, America would get cheap commodities (Copper, Aluminum, gold, steel, fruits, etc), and more jobs for the service and science/engineering sectors.
What exactly will they be gaining from the TPP, if anything? Well, you see, for these countries, exports are the way to go. So, having free access to other markets will end up paying off in the mid to long term. Some of these TPP countries have cheap labor (Mexico, Vietnam, Singapore). Others have advanced technology (Japan, Canada, Singapore). And others have wealth and resources/human capital (Australia, New Zealand, Canada). They would be cooperating among themselves, trading and exchanging goods and services among themselves, and gaining political and ideological muscle.
To put things in perspective, Australia wants to become a big player in the world stage; they already have the wealth to do so, they only lack the "presence" for it. Canada wants to become less dependent on U.S. politics, and rejuvenate its demographics. Mexico wants to become less dependent on the U.S. as well as continuing its path towards developed status. Japan wants to protect its interests (Protection from China), wealth (factories in other countries), and regain political influence in East Asia. Vietnam wants to continue its economic growth, as well as become part of the "big boys club".
There's so MUCH more to it than what I have posted, though. Even some drama between these TPP nations. For example, this one time in 2011 (Maybe it was 2010), Japan bribed Peruvian, and Australian politicians to dissolve a ban on whale hunting, but then Australia told Mexico (Australia changed their mind at the last second, and Mexico was voting to dissolve the ban at first), and they both voted in favor of the ban leaving Peru looking like a corrupt bastard, and Japan as the "evil" country against animal wildlife. I believe Japan attempted to do the same the next year with another country, I believe. I can go on and on about this topic.
Anyways, this is a simplified and reductionist explanation. Like I said, I encourage you to learn more about it yourself. It's not as boring as it might seem. But yeah, overall, I would say Trump was the worst case scenario for American influence on these countries. There is a strong Anti-Trump sentiment in Latin America, for example. This might seem like a moot point since "lol Latin America. lol", but it adds to the growing list of countries not under America's control anymore. Australia has historically aligned with the U.K. and the U.S. , but if Australian exports are at risk you can count on Australian going off their "leash". Same goes for Japan and Canada. Like I said, take all of this as a brief summary; not the whole thing. This is an over simplification.