If the Democrats hadn't invoked the nuclear option in 2013, couldn't the Republicans could just invoke it today?
Yes, but they might not.
The only legal difference is that it is already done. So they don't have to go through the vote. And they don't have to justify it in court later. It's already been accepted as valid. That wasn't a huge hurdle, but it was a possible hurdle. Until someone tried it, it might not have worked. The threat of using it had value in and of itself.
There are still Republicans who favor the filibuster for its own sake. Some others have called for Republicans to remove the filibuster immediately so that Neil Gorsuch can enter with a simple majority and they can pass legislation alone. Yet they haven't done so. Three Democrats voted against the nuclear option in 2013. Three Republicans would have been enough to block it in 2017.
It's not evident that they have the votes to pass it. Three of the seven Gang of Fourteen Republicans are still in the Senate: Susan Collins; Lindsey Graham; John McCain. While they might vote for it eventually, they are unlikely to do so if they have more immediate alternatives.
Without it, it is likely that Democrats would have pressured Trump's appointees to have provided more documentation but would have eventually voted for most of them. Why? Because otherwise, Republicans would be able to use it.
The Republicans chose not to use the nuclear option in 2005. As a result, they had the filibuster for the first four years of the Obama administration. If the Democrats had not used it, Republicans would have had it for six years. And they'd remember the power of the filibuster. They might not have been so eager to eliminate it. It took the Democrats six years (including the last two years of the Bush administration).
We don't know what would have happened if the Democrats had not used the nuclear option in 2013. But we do know what happened when they did.