Yes.

As the majority, the Republicans in the Senate can unilaterally eliminate the filibuster rule, then pass anything they like. They don't need Democrats' votes at all. Republicans have already done this in the past when they wanted to ram through a bunch of judicial appointments and again when they wanted to get several Supreme Court justices through without any Democrat support.

The only thing stopping them is that the Republican party wants this shutdown to happen. It gives them another example to point to when they want to claim that government never works, license to harm government workers they don't like, an easy way to have government workers leave on their own, an excuse to fire government workers they want to get rid of, and another thing to blame on Democrats (because their base doesn't know any better).

They control all three branches of government. How is it anyone else's fault?

What have Republicans offered in negotiations to help unblock the logjam?

Republicans have 50% of votes in both the House and Senate, but they can’t convince even 20% of the most moderate Democrats in red states (to get the last 10% needed to pass a budget bill). Why is that?

I’m not American. Help me understand?

Are they all of a sudden powerless? "Oh noes! we control the judicial, executive, and legislative branches of government, but those meany liberals won't let us balance the budget" ?

They control the body. Their solution to not passing a budget, nor a continuing resolution was... to just go home.

You can negotiate a budget during a shutdown. You can't negotiate if you sent everyone home.