Is it possible to search the contents of Soviet newspapers without having to browse thousands of pages? Yes, sort of, for certain titles.
Letopis' gazetnykh statei ("Chronicle of newspaper articles") is an index to all substantial articles published in a number of Russian-language newspapers. It has been published since 1936, initially monthly, then, since 1977, weekly. It includes coverage of the following titles:
Other titles were also indexed throughout the publication's history, for example Московский строитель and Ленинское знамя (Петрозаводск) in the 1950s, Совхозная газета before it became defunct in 1954, and Книжное обозрение, Московские новости, Поиск, Правительственный вестник, Рабочая трибуна, as they emerged or grew in prominence during perestroika and glasnost.
Letopis' gazetnykh statei is organized by topic according to the Soviet bibliographic classification scheme, which underwent several changes over the years, though "Marxism-Leninism" always remained the first category. The basic outline of the classification scheme was typically included in each issue. Quarterly name and geographic indexes (указатели) were also published (in the 1980s included with each issue).
Letopis' gazetnykh statei is a powerful, though very clunky tool. If you are working with a short range of dates, it is absolutely indispensable. If you are tracing developments across many years or decades, working with its hundreds of issues becomes unwieldy and too laborious. Nevertheless, there is no substitute for it, and, criminally, no publicly available digital version as of 2022 (a subscription service exists in the Russian Federation: http://letopisi.dlibrary.org; UNC University Libraries is investigating it.)