The practical need for a $T$ or $t_{max}$ is based on the important intellectual technique of doubt and its association with cost containment. At a certain point, it is wise to abandon an objective the resolution of which cannot be certain. This is the minimization of a much wider loss function associated with the business into which the seeking of the objective fits.
Dismissing the easily retained knowledge of the state of the independent process variable $t$ in the context of its domain $[1, T]$ as irrelevant is like working on a project without project management. What things to skip and what risks to take cannot be accurately evaluated, reducing the probability of success in relation to its optimum.
Certainly the value gained at any step is an indicator of the advantage of the current path, an affirmation that impacts the probability of achieving favorable results before $t$ reaches $T$. Certainly any knowledge that can be used to predict possible values along the paths that begin with choices that can be made at any step should impact that choice.
The question author's intuition that the critic should be aware that the stakeholder may pull the plug on the project soon is correct. In terms of the mathematics, whether urgency can be formulated as $u(t) = \dfrac {T - t} {t}$ and used as a term or a factor requires some thought about the probabilities involved. The advantage expression should reflect the application of the basic relations of probability to the probability distributions as seen considering all the information known at any temporal point along the pursuit of the objective.
The only occasion when a variable that can be known should be excluded from the advantage expression is if it can be shown to be absolutely irrelevant or that its relevance is sufficiently small to warrant exclusion to gain computational speed in exchange for loss of accuracy and/or reliability.