I am not aware of clear direction that have been established to know the cause and effect. But provided the current knowledge, what you said seems likely. There are however few more factors to be considered. For instance the amount of auro-antibodies produced also goes up with ageing. There might be a few reasons for this - 1) The regulatory lymphocytes (TRegs, for instance) function declines with ageing. These are the cells that acts like police to mellow down things. A component of complement system such as c1q also interacts with mitochondria in deciding the fate of some adaptive immune cells - how will they respond to self-antigens. Since we know mitochondria gets fucked up in ageing, this may also be another axis for autoantibody production.

The third axis is more in relation to your question. It has been shown body can produce auto-antibodies against inflammatory cytokines. Their function is mostly to neutralize the cytokines, such as TNFalpha. While this might have some benefits in a long run - such as in chronic low grade systemic inflammation it may break the immune cells tolerance to self-antigens. This may exaggerate the autoimmune related inflammation at the site. And, hence act towards promotion of auto-immune diseases.

However, further studies are required to establish a clean mechanistic picture. As of now these are most likely hypotheses which can be considered. On the other hand autoimmune condition may promote inflammation and accelerate ageing. There are some suggestions floating around the idea of testing drugs given in auto-immune diseases for other ageing pathologies.