Drops the serde_eventsource dependency

serde_eventsource is taking a while to update to reqwest 13.*. This
PR implements handling SSE in the library code.

examples/generate-content-stream.rs

@@ -0,0 +1,32 @@
+use std::{env, error::Error};
+
+use google_genai::prelude::{Content, GeminiClient, GenerateContentRequest, Role};
+use tokio_stream::StreamExt;
+
+#[tokio::main]
+async fn main() -> Result<(), Box<dyn Error>> {
+    tracing_subscriber::fmt().init();
+    let _ = dotenvy::dotenv();
+
+    let api_key = env::var("GEMINI_API_KEY")?;
+    let gemini_client = GeminiClient::new(api_key);
+    let prompt = vec![
+        Content::builder()
+            .role(Role::User)
+            .add_text_part("What is the airspeed of an unladen swallow?")
+            .build(),
+    ];
+
+    let request = GenerateContentRequest::builder().contents(prompt).build();
+    let mut response_stream = gemini_client
+        .stream_generate_content(&request, "gemini-3-pro-preview")
+        .await?;
+
+    while let Some(content) = response_stream.next().await {
+        match content {
+            Ok(response) => println!("Response: {:?}", response.candidates[0].get_text().unwrap()),
+            Err(err) => eprintln!("Error: {}", err),
+        }
+    }
+    Ok(())
+}