The sources of outcome variability here include: that among individuals having the same treatment or control, that between treatment and control, that between control groups in different populations, and possible interactions between treatment and populations. What you propose is risky at best, as there could be systematic differences between your population and the populations evaluated in the studies from the literature.
A one-sample t-test evaluated against an overall fixed value would be particularly risky. It wouldn't take the variability in outcomes among populations in the published studies into account at all.
Even a simple 2-sample t-test as you seem to propose could be misleading. That would use the standard error of outcomes among individuals in your population and the standard error of outcomes among populations in literature studies for the comparison. Those aren't really the same thing.
What you might do is to evaluate whether your treatment-group outcomes are far beyond the distribution of control-group outcomes in the literature, based on the standard deviations among the populations. If so, you might be confident that your values represent a treatment effect instead of just a population with a somewhat extreme control-group value and no true treatment effect. The risk is that might be too stringent a requirement and you might not perform a study that could have been useful.
It would be best to get at least a pilot sample of control outcomes for your population.
