As a legal matter, I think Horning's latter allegation is better than the former. I will explain why in a bit.
Libertarians have long argued for an end to taxpayer-funded primaries. Their position is that parties are private organizations that should be able to pick their own candidates...and should do so at their own expense. Of course, the notion that political parties are completely private organization, free from governmental interference was rejected by the courts in a challenge to the "White Primaries" many state Democratic parties were holding in the South post-Civil War.
I strongly disagree with Libertarians on the elimination of primaries. As a Marion County Republican, I am very familiar with the local party leaders use of slating, a formal endorsement process, to control who gets the nomination. Although slating was supposed to be about party workers endorsing candidates, the process has become so rigged over the years, it essentially amounts to party bosses, not party workers, picking a slate of endorsed candidates. Fortunately, we do have a check on that...namely a primary election in which a candidate by can be nominated by the party electorate despite not receiving the endorsement.
The Libertarians want to take that critical protection for voters away to leave the nomination decision entirely in the hands of party bosses. Their argument is if the party bosses pick bad candidates, the public will punish them by choosing other candidates. This, Libertarians argue, will eventually, years down the road, lead to reform and better candidates being nominated. I don't buy it one bit. Locally, I see party leadership which would rather endorse candidates who have little chance of winning the general election than nominate candidates they can't control 100%. Taxpayer money spent on putting on primaries is money well spent as primaries provide often the only chance voters have to have meaningful input in terms of who is elected.
As far as legalities, I think Horning's argument that Libertarians, other third party candidates and independents have no say as far as election law enforcement is on solid footing. Republicans and Democrats on the Election Commission have no interest in protecting the interests of third party and independent candidates and often appear to be working in tandem in keeping non-major party candidates out of the elections. I have seen court decisions that indicate such arrangements, in which the two major parties carve out exclusive territory for themselves, are unconstitutional. Whether Horning can hurdle standing and other challenges to succeed on this claim though remains to be seen.
As far as the challenge to the primary system, Horning cites IC 3-10-1-2 as "creating an arbitrary threshold" that has to be met before political parties and independents can participate in primary elections. Undoubtedly Horning is referring to the 10% requirement in that statute. But a review of that statute reveals that it only says parties must hold a primary when they receive 10% of the vote in the Secretary of State's race. That statute doesn't seem to bar other parties from having taxpayer funded primaries when they are below the 10% vote requirement. Here is IC 3-10-1-2:
Each political party whose nominee received at least ten percent (10%) of the votes cast in the state for secretary of state at the last election shall hold a primary election under this chapter to select nominees to be voted for at the general election.In short, that statute would appear to require the Libertarians to have a primary if its candidate received 10% of the vote in a Secretary of State's race. But the statute does not bar Libertarians from participating in primaries now, when the party's Secretary of State candidate is getting less than 10%. Although many Libertarians are deadset against primaries, the Libertarian Party would be foolish to pass up the opportunity to participate. Primaries would give Libertarian candidates additional publicity and help generate interest in the party. It would also give Libertarian-leaning voters the opportunity to so declare by participating in the primary.
Possibly there is another statute that prohibits Libertarians from participating in primaries. I have not spent a lot of time reviewing the state's poorly organized election code to find it.