No, it doesn't. There is no such thing as an "open shop" in the US, even under right to freeload. Union membership is determined by job classification. If you are working in a unionized environment, and your job title is a union job title, you are in the union. You will be covered by the contract your union bargains with the employer. The only question, then, is whether you pay the dues to cover the costs of union membership. In a right to freeload state, you can choose not to pay the costs.
You both have it mostly backwards. The Taft-Hartley act mostly bans closed shops and allows states to pass “right to work” laws making payment of any agency fees to unions optional.
In no state are any employees compelled to formally join a union. In an American-style “closed shops” in a non-right-to-work state, if you don’t join the union the bargaining agreement can charge nonmembers an agency fee representing most of the dues they would pay as members. Not to mention pressure tactics to join.
Also especially in the trades in major markets unions control apprenticeship programs and hiring halls and make it virtually impossible for nonmembers to get union work, as well as gatekeeping access to apprenticeship programs in the first place.