Learn English – Will I be missing

Those two sentences are saying two completely different things:

Everyone gets to sing one or two songs, but sometimes our manager will sing for ages by himself until he runs out of steam.

This one is saying that, occasionally, the manager sings for a long time. It does not actually imply that this always occurs... the word sometimes is telling you that.

Everyone gets to sing one or two songs, because our manager will always sing for ages by himself until he runs out of steam.

This one is saying that the manager always sings for a long time and the because makes the first clause beg for the word only:

Everyone only gets to sing one or two songs,...

If you really want to simplify the sentence and keep the same meaning, the simplest thing to do is to remove the word will:

Everyone gets to sing one or two songs, but sometimes our manager sings for ages by himself until he runs out of steam.

From what you added in the comments, this may be a good way to word it, with some slight changes for what you're trying to say:

Everyone usually gets to sing one or two songs but sometimes our manager will sing (or sings) for ages by himself until he runs out of steam. When that happens, most of us only get to sing one song.

There are many ways to say the last part of the phrase based on what you're actually trying to say. You could say:

When that happens, most of us don't get to sing at all.

When that happens, we're lucky if we can sing even one song.

Just a couple of options in addition to what's in the example above.

The only context these might differ is in a more complex sentence, such as one in which "where" might not refer to a physical place. Otherwise, you are correct, these have the exact same meaning, and do in the sentences you have provided.

As a possible example of an exception:

I wrote an article about playing basketball where I discussed the potential health benefits of the sport.

Note this would have substantially different meaning than:

I wrote an article about playing basketball at a place where one could remain outside of direct sunlight.

In the first, where refers to the article written. In the second, where refers to the place one is playing the sport. In the second case, at a place helps clarify the meaning.