I think even the author of this paper would agree that its title is simplistic (although certainly likely to generate media interest).
Firstly, the answer depends on which animal products and plant-based alternatives we are discussing. If comparing a highly processed, high fat red meat with an unprocessed legume, then I would agree with the title. However, if comparing milk and oat milk, the nutritional and health benefits are clearly better from milk, and the environmental aspect dependent on choice of functional unit (10.3389/fnut.2022.957486). And in other cases (e.g. eggs, fish), functional alternatives are, in most parts of the world, not available.
Secondly, the quantification of healthiness is challenging. From a nutritional perspective, the animal and the alternative have quite different content and bioavailability (10.1071/AN21235). From a public health impact perspective, I would agree with the title in most cases, but not for example in vulnerable populations in LMICs, where low diversity plant-dominant diets are a driver of malnutrition.
Environmentally, it really depends upon your functional unit. Should we compare carbon, water, land use, or biodiversity? And per kg, per gram protein, per $? In most cases the author's assertion will be correct, but not in all (10.4060/cb8054en).
In general, it is better to compare diets rather than individual foods and understand that there will not be a simple X is better than Y answer. There will always be trade-offs.

I think that the only suggestion for maintaing good health is not a plant-based diet but a diet without processed or supprocessed foods...

Plant-based foods are often considered healthier than meat because they  have low saturated fat and cholesterol with rich fibres, antioxidants and essential nutrients that support heart health and the digestive system. Diets rich in whole plant foods such as vegetables, legumes, fruits, and whole grains have been reported to lower the risks of chronic diseases such as diabetes, certain cancers and heart disorders. 

Plant based food products are healthier than meat. Because they are rich in vitamins, Fiber's, Anti-oxidants and other essential amino acids like cysteine, cystine and methionine that are link with lower cholesterol and lower risk to diabetes, heart attacked, obesity and certain cancer. but in case of meat provide complete protein, B12 and other essential nutrients. Ultimately, a balanced diet which including plant and animal based products are healthiest and most sustainable choice.