Browsers will execute inline JavaScript
All browsers will execute inline JavaScript provided they have JavaScript enabled. Whoever told you this was wrong.
Browsers will likely continue to execute inline JavaScript by default.
Inlining JavaScript is the easiest way to make a piece of script occur at a particular point in the page. The internet is democratic. You don't need to be a computer scientist to hack together a renderable HTML page with some blinking text and a dancing penguin. This is how the web is supposed to be.
Also, sometimes it is useful to be able to pass through a content managed JSON object from the HTML to a static script. Any browser that removed this would become instantly less useful, and people would migrate away.
Issues with inline JavaScript (why this is actually a good idea)
Allowing Inline JavaScript makes cross site scripting (XSS) attacks fairly easy. The attacker enters some JavaScript into a web form, perhaps a comments box, and the server then renders that script to the page. The script can then do things like steal login credentials or redirect to another page containing malware.
Currently XSS must be dealt with on a per server basis, and it's actually harder than you might think as there are many ways to execute script. Implementing a simple header element that turns off inline script would be a much easier way to guard against all XSS.
It's best practice not to use inline JavaScript if possible
You should think twice about using inline JavaScript. Separation of concerns (HTML5 for meaning, CSS3 for styling, JavaScript for behavior) remains good practice. It's tidier and easier to maintain. Also, by separating your JavaScript into a separate file, you gain the benefits of caching. The script will not need to be downloaded each time a page on your site is viewed.
Optimising for pure speed
The exception to this is if you are optimising for speed. Placing your script inline at the end of your file will ensure your content is visible as soon as possible. This is a technique that Google are fond of. I don't hold with it personally as it makes your code messy, but it will render the page content slightly faster.