I think that biodiversity and ecosystem conservation are tremendously important fields and deserve much more attention from the EA community as a whole, so I am very happy to see someone else being interested in them as well!
(To those wondering why these fields are important: Biodiversity is an important protection against zoonoses and a big cornerstone of general global biosphere and ecosystem service resilience. Various ecosystems provide services directly or indirectly to humans, like pollinators or the Amazon basin which sustains big parts of the water cycle for South America. Biodiversity risks make these systems much less efficient and/or effective or may ultimately lead to ecosystem service failure or collapse. Species are interdependent in complex and sometimes unforeseen ways; chain reactions when certain species' populations become too small or locally extinct can ripple out transitively to damage the entire ecosystem. Long story short: These ecosystems provide services to humans which we cannot live without, such as pollination, oxygen supply, food supply, pollution filtering, and even water supply. Our planet is, literally, one big ecosystem made of many smaller ecosystems and biodiversity ensures their resilience and functioning.)
Alas, it is a very complicated field which is interdisciplinary with many others (like classical conservation, biology, economics, tourism, politics/policy/advocacy and even criminology). Also take into account that while biodiversity and ecosystem conservation are closely related, they are not completely identical. The people working on these topics come from different backgrounds and hence sometimes have very different perspectives and strategies (which is both good and bad). Ecosystems can be under threat through secondary motives (i.e., they fall prey as "collateral damage" to other human activities) while biodiversity threats can happen due to primary motives (for example, poaching targeting important species which, when lost, have significant negative impact on their whole ecosystem).
I will list some angles I know for tackling this set of issues. This is but a very incomplete list because, as I mentioned, it is a very complicated field that touches on many disciplines:
- Community-based conservation, in which local communities are involved in conservation efforts. This must be set up in such a way that short-term overexploitation is not profitable but only sustainable activity is.
- Degrowth & ethical population reduction: I personally think this is one of the most effective ways to go. Human activity (settlements or agriculture) by now covers more than 60% of the global land mass and is projected to cover about 90% by 2055. Almost all this activity leads to ecosystem degradation or full destruction. Simply leaving bigger parts of the planet alone is the best chance we have. To my personal dismay, this approach is not viewed very favourably within EA (or globally, for that matter), due to the widespread belief that growth is inherently good and/or required, and that reducing human population size is bad. (I recently wrote a post about it which was mostly met with disagreement. I tried to start conversations with all commenters, but unfortunately none replied back.)
- Plant-based diets: As global agricultural land use is dominated directly and indirectly by meat production and continues to be so even more, plant-based diets would free up land (e.g. for rewilding efforts) and slow down ecosystem destruction.
- Tackling climate change: As you already pointed out, global warming has significant negative effects on many ecosystems as well. I would argue, though, that one should focus on other angles if you want to protect ecosystems and biodiversity because climate change already receives a lot of attention (though still not enough), while many other angles are severely neglected.
- Rewilding efforts: Unused land is converted back to wilderness areas.
- Sustainable agriculture: Instead of making agriculture as efficient as possible, one can alter agricultural practices in such a way that many species can coexist on agricultural land with only minimal efficiency losses. Corridors, wildlife bridges and "wild islands" are some examples. Agriculture should be "organic" so as not to damage other species as collateral damage when trying to control "pests".
- Politics, policy and diplomacy: Globally, many countries do not assign great priority to any conservation & biodiversity efforts. Environmental crime, pollution, overstepping of quotas etc, where even defined in national laws, is seldom properly prosecuted. If prosecuted, fines are ridiculously low and it is economically rational to violate laws and regulations, even if caught. The process behind the IUCN Red List appendix augmentations has more to do with different countries supporting each others' exploitation efforts rather than actual conservation efforts. Generally in current politics, ecosystem health is almost never prioritized over short-term economic gains.
- Strengthening law enforcement and prosecution possibilities: At the national level, law enforcement for anything related to the environment is underfunded, understaffed, undertrained, underequipped and underpaid. Internationally (like with a lot of international law), enforcement instruments are lacking.
- Anti-corruption: In many countries, conservation efforts are greatly hindered by prevalent corruption.
- Economics & poverty: Due to international differences (economic & legal risk differences, affluence differences), it often pays off to damage ecosystems. Providing alternative sources of income or wealth would be important. Damaging ecosystems should not be the most rational course of action for people that want to achieve a middle-class standard of living for themselves and their dependents.
- Education, advocacy and awareness: As you noticed yourself, even within EA the topic is almost unheard of. There is a big gap in knowledge, presumably because it is so multi-dimensional and complicated, because - much like with climate change - there are "long" (by human lifetime standards, not by planetary standards) delays in negative effects, and because there are big economic interests (much like fossil fuel companies with climate change) that want to keep biodiversity and ecosystem risks out of public perception. Thus, awareness and education work - even promoting these topics within EA - might be a very good idea. School education materials, online campaigns, conventional ads in countries with very low awareness levels, promotion of research, advocacy, contacting politicians, writing information material which points out the dangers of global ecosystems and biosphere degradation/collapse...
- Retreat and reserve zones, corridors: Declaring certain areas of ecosystems as strictly off-limits for any human activity has proven to be effective for certain ecosystems. Species can retreat there, reproduce, and when their numbers are replenished and they run out of space, "surplus" specimen spill over to adjacent areas which are open to economic use. This approach however requires good geographical distribution of such reserve zones: in terms of spot selection, evenness of distribution, corridor connection, and generally declaring larger areas as such reserves. Corridors ensure that species can spread over bigger areas.
- Funding: Many smaller organizations lack funding. Government budgets are way too small.
- Many families or ecosystems face specific threats, like herbicides and pesticides in agriculture which damage almost all insect populations, or certain kinds of pollution. Regulation/banning of certain products or practices would be the way to go.
Hope this list helps. There are many more interventions, I only listed the ones that came to my mind right now.
A great book that might give you a glimpse of the complexities in this field is "The Extinction Market" by Vanda Felbab-Brown. She focuses on wildlife crime, but some of the interventions are transferable. The reason why this book is a real treasure, though, is that she does a great job of explaining the advantages and disadvantages of different interventions and how they tie together with other considerations (like economical and political ones).
There is also an "Effective Environmentalism" group on Facebook, you might want to repeat your question there.