One specific development which has been accused of putting up a barrier to third-party participation in presidential elections in the US, is the management of the televised debates.
These were which went from being controlled the League of Women Voters, which was non-partisan, to being controlled by the Commission on Presidential Debates, which while nominally non-partisan, is in practice bi-partisan, as it was founded jointly by the chairs of the RNC and DNC.
The CPD has been criticized for effectively setting the bar for participation such that third parties are prevented from getting the recognition that would allow the public to learn about them and enable their growth.
Attempts by third parties to challenge this system in court, so far, have failed.
Regarding State and Local politics:
It may vary by state, but at least here in NYS, it is actually fairly easy for a third-party candidate to get their name on a state and local ballot. The substantial barrier is applying on time with a modest number of signatures, which a few hundred volunteers can gather with a focused effort, in about a month. I have participated in this kind of effort, and would say, anecdotally, that nearly half of voters are supportive, in principle, of the idea of having more choices at the ballot, even if they might not go on to vote for the minor party.