This has been tried, and it failed. The primary problem was that startup time of embedded Java applets was abysmally slow. You'd watch the "java loading..." progress bar in the browser window and wait for several seconds before it would finally start. Flash may have gotten its name because it had an incredibly faster startup time than Java.
Java itself is a very open standard (OpenJDK, Java language standard, etc.). Openness of the standard wasn't the problem. In fact, it offered a much stronger uniformity of experience no matter what browser you used, unlike web technologies of the time.
The security issues with Java and Flash were the final nail in the coffin, but they had already been fading -- Java due to UX issues and startup lag, Flash due to it simply being a divergent technology from the web itself.
Java itself is a very open standard (OpenJDK, Java language standard, etc.). Openness of the standard wasn't the problem. In fact, it offered a much stronger uniformity of experience no matter what browser you used, unlike web technologies of the time.
The security issues with Java and Flash were the final nail in the coffin, but they had already been fading -- Java due to UX issues and startup lag, Flash due to it simply being a divergent technology from the web itself.