No, the reason the "there is always an in-party Senate spoiler" effect (when they have a Senate majority) seems to be more true of Democrats is because it is more true of Democrats, and the reason is that when the two parties in rough balance by popular support (or even rough balance in Presidential electoral prospects, which has the same directional bias as the Senate but of lesser magnitude), the Republican Party has a systematic edge in dominance of states, which translates into a systematic advantage in the Senate, which means that when the Democrats have a Senate majority, it tends to have a decisive segment in red-state Democratic Senators who are unreliable on key priorities.

The issue being discussed in the Senate is not a symmetric issue resulting from near balance in support between the parties.

Equal in voting rights. Gerrymandering has been perfected by Republicans. Through that they manage to dilute votes of the opposition. Other measures discourage voters likely to vote against them, like people who cannot easily take time off to vote in person or who have changed their name. Blocking rank choice and maintaining first past the post also disenfranchise third parties, and reinforces the power of incumbents.

Trump himself admitted it's better for Republicans when fewer people vote.

Argh. Too late to edit. Something else outside first past the post* like ranked choice voting.