This is a complicated question, as there are likely few people (I am not one of them) who are fully knowledgeable in the scope of both DVM and MD or DO (which is the other type of physician degree) in combination with PhD, and I do not know if someone with that knowledge will see this question. But your question is a very good question - I would recommend you seek out and ask MD PhDs, DVM PhDs, MDs, and PhDs to explore your options. It's an important decision.
I can help from the perspective of an MD/MPH (Masters in Public Health) who has done some research, and I have colleagues who do many different types of research with varying combinations of degrees. MD or DO alone, MD or DO + PhD, or just PhD (e.g. in fields like biomedical engineering).
There are a number of MD-PhD or DO-PhD combined programs where usually you do 2 years of med school, then complete the PhD (which often is 4 but can be much more), then return for the last 2 years of medical school - and may or may not go on to complete residency. Some MD/DO PhDs practice clinical medicine, whether they do research or not. I imagine this is much the same for DVM-PhD programs.
But more to the point, what is most important is for you to reflect on your ultimate goals. Are you specifically interested in working on animal models? Or do you want to obtain the education that provides you the most possible options, so that you can determine along the way where you feel you can make the greatest impact?
There are vast numbers of ways to engage in research that improves the health of humans, and animal models is only a subset of the diverse possibilities.
The potential advantages of DVM over an MD/DO is something I had not previously considered, and I can see how expertise in the animal models would be beneficial in some research. But again, are you sure you want to focus primarily on animal models? Consider that DVM may limit your scope of what you can do in clinical research involving human subjects.
The other paths include MD or DO alone, MD PhD, DO PhD, PhD alone (in fields like Biomedical Engineering etc), PharmD (pharmacy doctorate), or others.
Of the two you mentioned, the MD-PhD (or PhD alone) may open more doors for you in research overall, since animal research is only one section of medical research. The MD/DO when combined with PhD does give advantages over PhD alone in some circumstances, from what I understand, but is not always necessary.
What an MD/DO can give you is a thorough grounding in both the fundamentals of human physiology/pathophysiology AND its clinical management. Completing a residency in clinical medicine also gives you the experience and skills to actually practice medicine in addition to doing research.
Again, I recommend talking with people with the degrees and combinations I mentioned. Your school's advising counselors should be able to help arrange that, or contact a local university with a med school or phd program to request to speak with their advisors, or with researchers. They may help you as a prospective student.
As a parting thought, you might also consider public health if you are aiming to do what you say here:
see human suffering diminished and eliminated as much as possbile during my lifetime.
Public health is sometimes overlooked, since people don't necessarily know about it as a field - but they definitely know the effects of it. Public health is all about improving the health of entire populations through multiple angles, including many branches of research. It incorporates (and offers specializations in) epidemiology, biostatistics, disease prevention, education, program design and analysis, population management, policy and administration, and other elements. Vaccinating the population has been a joint effort of medical research and public health.
Read about public health and maybe talk with the department at a university - it's actually a really rich and diverse field with lots of opportunities.