You've hit on a key insight. It absolutely matters *where* on Earth you launch from, and the choice involves a complex trade-off between that "free boost" and other critical factors. The southeastern coast of the United States (Cape Canaveral, Florida) was chosen as the optimal compromise for several powerful reasons.

While the rotational boost is a huge factor, it's not the only one. Here’s the breakdown of why the location is so specific:

### 1. Maximizing the Rotational Boost (The "Why South")

As you correctly noted, the rotational velocity of the Earth is greatest at the equator and decreases to zero at the poles.
*   **Equatorial Speed:** ~1040 mph
*   **Cape Canaveral (28° N):** ~910 mph
*   **A launch site in Maine (45° N):** ~740 mph

**So, why not just launch from the equator?** The U.S. territory of Guam is much closer to the equator, for example. This is where the other factors come in.

### 2. The Overwhelming Safety Requirement: The "Overwater Downrange"

This is arguably the most important factor after the boost. If a rocket explodes, or if a first stage separates and falls back to Earth, you want it to come down in an unpopulated area: the ocean.
*   **Cape Canaveral is on the East Coast:** Launching eastward sends the rocket's path immediately over **2,000+ miles of open Atlantic Ocean** all the way to Africa. This provides a huge safety buffer for falling debris.
*   **The West Coast Problem:** Launching east from California would send rockets over densely populated southern California and then the entire continental United States. This is a non-starter for safety reasons. (Vandenberg Space Force Base in California is used for launches into polar orbits, which go south over the open Pacific, precisely to avoid this issue).

### 3. Reaching Desired Orbital Inclinations

The "inclination" of an orbit is the angle it makes with the equator. It's easiest and most fuel-efficient to launch into an orbit that matches the latitude of your launch site.
*   **Cape Canaveral's latitude (28° N)** is perfect for launching to the **International Space Station**, which has an inclination of 51.6°. Reaching this from the Cape requires a minor and fuel-efficient maneuver.
*   **Equatorial Launch Sites** (like French Guiana) are ideal for launching satellites into **geostationary orbit**, which lies directly above the equator. This is why so many communication satellites launch from there.

### 4. Proximity to the Equator vs. Logistics

While an equatorial launch site offers the maximum boost, it must be balanced with:
*   **Infrastructure:** Building and maintaining a massive spaceport requires a stable government, a skilled workforce, and immense industrial support. Cape Canaveral is in the continental U.S., with easy access to all these things.
*   **Transportation:** Rockets and their components are huge. Shipping them from factories in Alabama or Louisiana to Florida is far easier and cheaper than shipping them to a remote island.

### The Perfect Compromise: Cape Canaveral

So, Florida's eastern coast represents the **sweet spot**:
1.  **Significant Rotational Boost:** It's the southernmost point in the contiguous United States, maximizing the free velocity.
2.  **Perfect Downrange Safety:** The entire eastern trajectory is over open ocean.
3.  **Favorable for Key Orbits:** Its latitude is well-suited for reaching the ISS and other common orbits.
4.  **Superb Infrastructure:** It's easily accessible and has been built up over decades into the world's premier spaceport complex.

In conclusion, you are right that any eastward launch harnesses the Earth's rotation. But to get the most out of it *safely* and *efficiently*, you want to be as far south as possible on an eastern coastline. That’s why the U.S. uses Florida, Europe uses French Guiana, and other nations like India (Sriharikota Island) and Japan (Tanegashima Island) also utilize southern, east-facing coastlines. It’s all about harnessing that free merry-go-round spin and then dropping the used parts safely into the ocean.