I'll try to identify the key questions as follows. By my interpretation, your two main paragraphs can be summarised as: Just how big is the Rebellion for this to be called the Galactic Civil War? Why does Yavin sound like it's their only base? How significant a threat are they for the Empire to put so much priority into their destruction?
Your final paragraph adds more questions: Why weren't the Rebellion evacuating Yavin? What would the destruction of Yavin mean? What would become of the Rebellion if the Empire won that day?
Just how big is the Rebellion for this to be called the Galactic Civil War?
The truth of it is that although it's indeed a civil war, there isn't much of a conventional war going on at all. The Alliance to Restore the Republic wasn't the Confederacy of Independent Systems. The Galactic Civil War wasn't the Clone Wars. In the Clone Wars, a large number of member systems who left the Republic formed their own faction and eventually went to war with the Republic. In the Galactic Civil War, an underground resistance movement was fighting the Empire's oppression, and all of their supporters were doing so in secret while publicly remaining members and subjects of the Empire.
It is a civil war because it is raging across the galaxy - Rebel scum are everywhere when they are attacking, but also nowhere when the Empire wants to attack. That's the kind of war the Empire was fighting - guerilla resistance.
Why does Yavin sound like it's their only base?
This requires a bit of out-of-universe understanding. At the time of the Original Trilogy, the concept of irregular warfare and decentralised operations is still new and almost unheard of. That kind of thing belongs to the 21st century, not the 1970s and 80s. The Rebellion is by design a centralised professional core extending into a vast network of underground, irregular cells because George Lucas couldn't imagine it any other way, simple as that.
In-universe...well, I can't give a good answer to that, unless we go into Legends. In Legends, the three founding members of the Rebellion were divided on what strategy to adopt. One wishes to go into all-out conventional war in the style of the Clone Wars, the other wishes to go into full guerilla mode, focusing on harassment while building up resources, and the third wants to strike a balance. The all-out war guy eventually left the Rebellion to form his own, so that's how we ended up with the Rebellion we see today.
Yavin wasn't actually their only base. As I've said, the Rebellion was constantly on the move to evade attention and so avoid being forced into a pitched battle which they can't afford to fight if they can help it. Dantooine was a former base which they abandoned, eventually settling in Yavin. Later on, they moved to Hoth. It's likely the Rebellion relied on one hidden base as HQ and a bunch of other bases as outposts, depots etc. due to its centralised organisation. Wookiepedia cites the movie/novelisation as calling the Yavin base "Base 1", so if this is true then it could suggest the true strategic and political value of Yavin's base.
How significant a threat are they for the Empire to put so much priority into their destruction?
In truth, the Rebellion had always been at a disadvantage. If they engaged in conventional warfare, the Empire would triumph despite their quantity-over-quality military philosophy, for the simple fact that the Empire could afford to fight a war of attrition. Losing 50 TIE Fighters and pilots hurts the Empire less than losing 5 X-Wings and pilots would the Rebellion. Notice how Yavin was their first major victory. In real life, that's usually called "the turning point of the war".
So the Rebellion has always been a guerilla movement: Constantly mobile, never in one place for long, never facing the Empire head-on if they can help it, committing their precious military resources carefully to surgical strikes with carefully balanced risk-to-reward. They can't afford to lose their ships, but they will do it if they think it's really worth the risk or if they can't avoid it. Yavin, Hoth and Endor are examples of what happens when they go at it. Heavy losses, and when they do win it's only because the Force is with them. If we go by conventional analysis, they are never the winning side.
The Rebellion isn't that big compared to the Empire: If the Rebellion and the Empire put all of their conventional forces into one battle, it'll truly be the shortest offensive of all time.
So why does the Empire want to crush them so bad? Because they need to set an example: The continued existence of the Rebellion is a sign of Imperial weakness, of Imperial failure to keep the Empire's subjects in line. They would inspire others to join their cause. The Empire must crush the Rebellion while it is still weak without mercy. Left to fester and grow, the Rebellion will eventually become strong enough to engage the Empire in all-out conventional war in the scale of the Clone Wars - a scenario that the Empire may not survive a second time. Additionally, crushing the Rebellion will send a message to other would-be rebels of the futility of such thoughts, reinforcing in them to submit to Imperial domination.
The value of the Rebellion's destruction is mainly political.
Why weren't the Rebellion evacuating Yavin?
I can't remember which of the following two is the situation at the time:
- They were evacuating. That's why the main fleet isn't there. They evacuated. What's left on Yavin are the essential staff to prepare for the upcoming battle. Given the stakes, it's a battle they can't afford to run away from. They need to destroy the Death Star. Those that evacuated is the insurance, the back-up plan for the Rebellion's survival in case the battle was lost. Leadership figures like General Dadonna was probably there to direct the battle on the front. Notice that the political leader, Mon Mothma, was absent.
- They can't evacuate. The main fleet isn't present at the time. So they didn't have the ships to conduct evacuation with. Calling the fleet in is too dangerous considering the approaching enemy.
What would the destruction of Yavin mean?
Reasons as mentioned earlier: As Rebellion HQ, the destruction of Yavin is a political victory with regards to the imminent crushing of the Rebellion. It sends a message to the galaxy and to the remnants of the Rebellion.
What would become of the Rebellion if the Empire won that day?
The base on Yavin was most definitely not the Rebellion's entire existence. But defeat at Yavin would be equivalent to defeat on Hoth but with none of the ships escaping successfully. The entire Rebellion's efforts towards victory would be set back by years.
The consequences are many, but I would focus on just one: Defeat at Yavin would mean the loss of both Luke and Leia Skywalker. Unknown to the Rebellion, Emperor Palpatine is Darth Sidious. With the death of the Skywalker twins, Obi-Wan and Yoda's plans, as well as the prophecy of Anakin Skywalker could all be lost. The extermination of the Jedi could have been assured as a result. Can the Rebellion defeat, by Darth Bane's standard, the most powerful Sith Lord in a thousand years without a single Jedi?