Hi. I'm Pole. "Budżet" is pronounced with both short and long ż, depending on person and region. In most regions of Poland there is no correlation between written ż/rz and its pronounciation. This is one of the pains every polish kid has to go through and we just have to learn arbitrary rules and exceptions to where we write ż an where rz as there is no help from pronounciation. Same goes for u/ó and h/ch which also have identical pronounciation. Historically, there were differences in pronounciations but these were lost over time. There is a movement to completely eradicate these different forms to make the language a little bit easier, especially for kids. I think the time spent on these arbitrary, unnecessary details that cannot even be heard and very rarely providing any useful information could be used on learning something with a bit more value than just sentiment.