Does combustion occur more easily at higher oxygen concentrations or just higher oxygen partial pressure?
When you're dealing with questions like "how well does this burn", you are intrinsically dealing with the reaction rate of the combustion reaction.

When looking at chemical reaction rates, they are typically proportional to the "chemical activity" of the species reacting. (There's complications due to the stoichiometry of the reaction, but those corrections are confined solely due to the nature (formula) of the reaction, rather than any particular concentration/pressure conditions.) The definition of chemical activity can get complex in certain cases, but for most normal reaction conditions, the "chemical activity" of a reactant is proportional to its molar concentration - or in other words, how many molecules of the reactant are in a given volume.

For solution reactions, we typically use the concentration directly, but for gaseous reactants, we can take advantage of the ideal gas law and realize that at a fixed temperature, the concentration of a substance is proportional to it's partial pressure. Which means the chemical activity of a particular gaseous reactant will be proportional to its partial pressure.

So in determining the reaction rates of such things as combustion, it's the partial pressures of oxygen and other reacting gases that matter, rather than the total pressure. Absent any side reactions, or considerations such as thermal transfer, adding or removing an inert gas should not affect the rate of reactions.

(But note for combustion, which is highly exothermic, thermal transfer effects can't necessarily be neglected. The ability of the inert gas to either carry away excess heat - or act as a blanket to keep heat near the reaction point - could substantially change how quickly a fire burns in practice. But I'd expect it to be a complex relationship based on air circulation, convection considerations, etc.)