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Quasi-experimental design could work. They do that for other things. If they have a large enough sample size, they could hopefully counteract all those other variables with the number of subjects. I would imagine measures of feeling of well-being are somewhat subjective.
If those who report low sex frequency are more likely to misreport feelings of well being, for example, that could be a problem. and the same for those with high frequency.
One way to do this would be to take a very large random sample of people and ask sexual frequency and then track their report of a sense of well-being (or whatever depression-related variables are relevant) over time along with their sexual frequency. If those with low sex frequency tend to have emotional issues, then the experiment may require prescribing a certain frequency of sexual activity could be a way of doing it. They could have a group with sexless marriages (already, not prescribing it since that is unethical). Another group they could prescribe a low sexual frequency (along the lines of what they are doing already) an then prescribe another group with a high sexual frequency. If depressed people overreport sex in a categorical manner or something like that, this could be controlled for by asking the spouse for a report of sexual frequency. I don't know the research or the variables, but I think such a study is very doable.