They actually asked him multiple times IIRC for a right to make adaptations, and he refused them every single time.
That is incorrect. Tolkien sold the entire media rights -- film, stage, and merchandising -- for Lord of the Rings to United Artists in 1969 for just over £100,000; it is commonly claimed that this was because he was in need of immediate cash to pay overdue tax bills, but there was never any evidence for that. The cash amount is not nearly so small considering the economy of the day plus over half a century of inflation, and the deal also granted Tolkien a very substantial 7.5% royalty on all profits from any adaptations made.
United Artists inititally intended to immediately produce an LOTR movie, but ended up cancelling the project and selling most of the rights they had purchased to Saul Zaentz's film production company, which created a specialized subdivision -- Tolkien Enterprises, later re-named Middle-Earth Enterprises -- to handle licensing. United Artists retained, and continues to retain to this day, the global distribution rights for any and all film adaptations.
In 2012, the Tolkien Estate sued Middle-Earth Enterprises, Warner Brothers, and New Line Cinema over the production of LOTR licensed video games, claiming that the merchandising rights sold by JRR Tolkien included only the rights to "tangible" merchandising and that therefore they were only allowed to license the creation of video games that were distributed on physical disks. Moreover, the Tolkien estate claimed that Middle-Earth Enterprises was causing "irreperable harm to Tolkien's legacy" by allowing the licensing of LOTR-based gambling devices, i.e. the highly popular LOTR slot machines by WMS Gaming. Note that the irreperable harm claim was specifically about gambling, which the estate denounced as "morally questionable", and not any complaints about the content of storylines etc. That lawsuit -- and the counter-lawsuits filed by Warner Brothers and Zaentz Productions -- were settled out of court in 2017.