Will obligate aerobic bacteria die if placed in a buffer solution even in the presence of oxygen?
Answer
No. There is ample evidence that:

Most non-extremophilic bacteria grow over a broad range of external pH values, from 5.5 – 9.0, and maintain a cytoplasmic pH that lies within the narrow range of pH 7.4 – 7.8.

The explanation for this is another matter, but, before discussing that question there are two points of possible misunderstanding in the question that require a note of clarification.
Clarifications

What the electron transport chain establishes is an electrochemical proton gradient, or proton-motive force (PMF). This consists of two components§: the hydrogen ion concentration difference, ΔpH, and the electric potential gradient, Δψ.

The source referenced in the question shows hydrogen ions being pumped from the cytoplasm, across a membrane, to a space labelled “outside”, which, according to the figure legend represents hydrogen ions “pumped out of the cell”. This is incorrect. The hydrogen ions are pumped into the space between the inner and outer membranes of gram-negative bacteria, or the space between the inner membrane and cell wall of gram-positive bacteria. It cannot necessarily be assumed that this space has the same pH as the external milieu.


How is the proton motive force maintained?
A possible explanation of why a constant or, more striking, alkaline pH does not destroy the proton motive force relates to the two-component nature of the latter. Bacteria appear to be able to compensate for any decrease (or even inhibition) of either ΔpH or Δψ by a reciprocal increase in the other (Bakker and Mangerich,1961). One mechanism that illustrates how this can be achieved in certain cases is the use of an alternative terminal oxidase, cytochrome bd:

Cytochrome bd generates a PMF by transmembrane charge separation, but
does so without being a proton pump.

However there is a complex set of anion co-transporters involved in cytoplasmic pH homeostasis which may influence the membrane potential. The situation is neither simple nor, as yet, completely resolved, as far as I am aware (but it’s not my field).

§ The treatment of electric potential gradient, Δψ, tends to be downplayed in biochemistry text books. This may be because the generation of a hydrogen ion gradient is easier to understand than that of a membrane potential, or because the texts deal with the latter in sections on nervous tissue. There is a quantitative treatment of the free energy changes involved in Chapter 18 of Berg et al. online.