First of all, despite all the nonsense to the contrary, an Article 5 convention is not some revolutionary thing that would completely overturn our government. It is a part of our government. Its job would be proposing of amendments to the Constitution. You correctly state that the Republicans controlled the House and Senate in the 2000’s and failed to do anything. (Control of the Presidency and Supreme Court are actually irrelevant to the amendment process). That, however, is the whole point of the Article 5 convention. The option of the Article 5 convention exists precisely because the Founders realized that there might come a time when amendments were needed that would not be proposed by Congress because such amendments were contrary to the interests of the members of Congress.
Just an example, suppose it is determined that term limits for Congressmen and Senators are needed (I’m not stating that I agree with this; this is just an example). Certainly very few Congressmen or Senators would even think about proposing such an amendment. Why would they? They would essentially be voting themselves out of a cushy job. That’s what an Article 5 convention is for. Assuming that the public came to the widespread belief that term limits were a good idea, the state legislatures would petition Congress for an Article 5 convention. Said convention would then meet and propose the necessary amendment. Any amendment proposed by the convention would then be sent to the states for ratification in the same manner as amendments proposed by Congress.
The Republican party on the national level would have very little to do with such a convention. This method gives states essentially all the power. Of course, you might argue that the Republican party in the states is just as bad, if not worse than the national party. I would not necessarily argue with that. The remedy, then, would be abandoning the party and starting a new one. If that could be done first on a grassroots, local level, and built up to become powerful in enough states, an Article 5 convention could be successful.
If that were possible, an Article 5 convention wouldn't be necessary. We could simply have our elected representatives pass useful laws to run the country.