This claim is slightly confusing, as the title of the claim is "food does not have an impact on your mental wellbeing" however, the asker then provided information on how food does impact your mental wellbeing, so I am unsure what this poster is claiming, whether they believe it does or does not.
However, according to my research, food and diet does in fact have a great effect on mental health and wellbeing. Many of the neurons and activities in your brain rely on bacteria and sugars for example to be able to function. According to UCLA Health, “Serotonin is one of those neurotransmitters that is associated with mental health, and it’s produced by good bacteria. Up to 95% of serotonin is produced in the gut, so, we have to eat healthy foods to increase the presence of these good bacteria,” Dr. Gupta, a co-director of the UCLA Microbiome Lab, stated. This is just one example on how a major neurotransmitter is impacted directly by diet and health, as it is made in your intestines. The National Library of Medicine states that, "The effects of certain foods or dietary patterns on glycaemia, immune activation, and the gut microbiome may play a role in the relationships between food and mood", however they do also state that more research is required to show specific effects of different foods on the mental state and mind.
Now, this does not mean that diet is the sole factor in determining mental wellbeing, however it can attribute to either a positive or negative mental space. Of course, lack of neurotransmitters and mental disorders, i.e. genetic, trauma-based, etc., have the most impact on your mental state and what your mental health looks like, but the factor of diet is very prominent.
Both of these sources, UCLA Health and The National Library of Medicine are both reputable sources that have no need for any particular bias in this case, as they are not advertising anything or trying to convince readers one way or another, instead they state facts and findings about the body and mind relationship in humans. There are many blogs I came across stating these findings as well, however I find it best to find sources for matters like these directly from what is deemed as the most respected and credible sources, such as top universities and government programs that are believed to tell the truth to readers.
UCLA Health: You are what you eat: Diet may affect your mood and brain function | UCLA Health
National Library of Medicine: Food and mood: how do diet and nutrition affect mental wellbeing? - PMC