I don't have a complete answer but can provide some thoughts:
1) Adjustment does remove the confounding effect, but only if the underlying causal pathways are correctly specified. There are occasions where adjustment can cause bias rather than decreasing biases. For more information on this issue, search for collider bias and directed acyclic graphs.
2) Adjustment does remove the confounding effect, but only if the operationalization is correct. In other words, you have chosen the correct variable to represent the construct. There are multiple reasons why age may not be a good indicator of aging (the actual construct that is related to mortality.) For example, fatal heart disease can be lifestyle-related. It can also be related to immunological response and how body mediates inflammation. All these factors can have substantial difference within the same age. In the reporting side, under-report of one's age tends to go up with age, introducing some error that is correlated with age as well. If you control for age, and thinking you have controlled for age-related factors, chance is this assumption is usually over-ambitious. It's always more important to know what the control variables really means.
3) There are also other dynamics which can cause adjustment alone to be insufficient. For example, interaction between age and other variable(s) in the model can bias the estimate of age. Non-linear relationship between age and mortality can also cause simple adjustment for age alone an imperfect method.
My guess is in epidemiology, it's better to say "no" whenever someone asks if something can completely removed whatever... perhaps except "can randomized controlled trial completely remove biases?" Then "theoretically yes."