Other advanced ancient civilizations have relied on large river systems, for example:
Mesopotamian civilizations rely on the Euphrates River and the Tigris River.
Egyptian civilization relies on the Nile River.
Indian civilizations rely on the Indus River and the Ganges River.
Chinese civilizations rely on the Yangtze River, the Yellow River and the Pearl River.
In contrast:
Pella, the capital of the Macedonian kingdom, was situated on the non-navigable Vardar River.
Ambrakia, the capital of the Epirote kingdom, was situated on the barely-navigable Arachthos River.
Sparta was situated on the non-navigable Evrotas River.
Rome was situated on the barely-navigable Tiber River.
Etrurian cities were built in the basin of the barely-navigable Arno river.
Utica was built on the barely-navigable Medjerda River.
Tyre, Byblos, Syracuse, Athens, Carthage, Thebes and Knossos were not built near rivers.
It also makes me wonder why:
Ancient Greece's most powerful civilization didn't arise on the Haliacmon River or the Maritsa River.
Ancient Italy's most powerful civilization didn't arise on the Po River or the Adige River.
Ancient Phoenicia's most powerful civilization didn't arise on the Orontes River.
It seems like there are a few levels to your question if I'm understanding it correctly (and please let me know if I'm not):
I know #2 is the stated question, so I'll focus on that one, but just to address the others briefly-
What benefits did the rivers provide to Mesopotamia, the Nile region, and some civilizations in India and China?
These are not fully comprehensive, but to name a few key ones:
Safe, abundant drinking water: This is incredibly basic, but something that's necessary for human life, and allows for human settlement. If you go camping, one of the first things you want to locate for a campsite is a freshwater source, ideally with cleaner running water. Similarly, you didn't see many ancient human settlements arise with large populations in the middle of the desert or where there wasn't a consistent freshwater source.
Fertile land for agriculture- Particularly in the cases of the Nile and Mesopotamia, civilizations began to flourish because of the (loosely) predictable seasonal flooding from the rivers which brought both water and fertile soil (from the silt carried by the floods) onto the land. The populations which began to grow in the areas developed means of irrigation to divert that water to agricultural farmland.
Trade routes- Trade is a key part of a civilization's growth which can allow for much more effective use or expansion of its resources or can reduce the need for the presence of natural resources. The rivers in each example you provided were a key route for commerce. It's worth noting that trade is a network effect, which relies on connections to other populations and becomes more and more important over time while other civilizations develop. This is important in addressing the second question.
Given all this, the rivers clearly provided benefits to the civilizations you've listed, so How did other ancient civilizations arise without the advantages that large river systems provide?
I think a key point to note is that major rivers do not have a monopoly on all of those resources. Freshwater is an absolute essential for ancient civilizations, but it's abundant also in lakes, underground aquifers, and smaller streams and rivers. And even when it wasn't as easily accessible some ancient civilizations devised some fantastic means for getting it from one place to another, such as the ancient roman aqueducts that you still see across Europe. Similarly, fertile soil is not only found around river valleys, and while ancient Rome or many Greek city-states were in areas that were not as ideal for say rice or barley, the different soils still allowed for different crops to be grown. In addition, several of the cities you mention, while not being on the largest or most navigable rivers, still had the benefits of fertile soil from those rivers.
Admittedly, agricultural productivity was not a strength of all the civilizations you mentioned, much less all ancient civilizations, so there was a reliance on the third point- Trade. Like in a game of Catan, if you need a resource that isn't directly available from your geography, you can trade for it using another resource, and proximity to a port is a major benefit in that approach. It's worth noting that most of the civilizations you give as examples not lying on major rivers had the Mediterranean Sea in common (or Macedonia the Aegean). It looks like one of the deleted posts may have mentioned this, but seas with accessible port areas provide a great venue for commerce and generally provide even more trade links than a river ever could. Sticking with the game analogy, even when a particular civilization lacked specific resources, trade could be used as a wild card to obtain them, and as other civilizations popped up and grew over time, so did the value of trade access in relation to any other specific resource. In fact, if you focus on a timeline of when all the civilizations you mentioned first appear or expand, you'll notice that for the most part, those that appear later are the ones that benefited even more or were more reliant on trade. The Greek cities, Pella, Rome, and Utica all developed extensive trade networks, and Rome in particular grew well beyond the bounds of what its local resources would allow for, due to imports from its many colonies and metropoles.
Why did alternative or more powerful civilizations to those that existed not arise along nearby rivers?
The third part of your question I can't answer as clearly without specific knowledge of the populations around the river regions you mentioned, so maybe someone else will be able to step up there, but in general I would say explaining why a theoretical didn't occur in history is much more difficult than discussing some factors that contributed to what did occur. One answer to your question may be a matter of primacy- once Pella, Rome, Athens, Sparta, etc. gained enough strength in their respective locales, they simply dominated others which began to develop around them. Yes, empires and civilizations rise and fall on their own or while confronting others (and they did!), but while on the rise, each civilization would often expand to take advantage of nearby resources rather than competing with them. As I mentioned, once Rome had conquered the surrounding area, it could simply import the resources of its territories instead of competing with them, and so they enriched the empire rather than beginning their own.
I'll admit (and it might be apparent from the examples) that I have much less knowledge of the Indian and Chinese river valley civilizations, so would certainly appreciate if anyone can address those more specifically, but otherwise I'm happy to share sources or further reading on specifics. If you're interested in Mediterranean civilizations from the examples you gave, I would strongly recommend John J. Norwich's 'The Middle Sea' as a very digestible and expansive survey of the many empires and cultures that developed along its shores. It doesn't get as in depth on specific civilizations, but is a good place to start.