Suggestions for solvents capable of efficiently performing alpha-bromination and nucleophilic substitutions
As a general comment published procedures have been optimised. I don't have access to the paper you link so I don't know if the procedure is from process chemistry (where you expect a high degree of optimisation) or from med chem (where the priority is to get the compound out fast but there is still some optimisation).

But your questions are still valid. So I agree that doing the bromination - amination in one pot using the same solvent is a reasonable experiment to try and actually acetonitrile is a good choice. Why do you think acetonitrile is nucleophilic? it is not (unless you deprotonate it with a strong base). So it is very possible that the first researcher tried your approach initially and possibly found that the bromination was not very selective or the amination step was too slow, or low yielding and switched to the current procedure.

I also agree that you can use a base instead of excess of amine, for example an amine base like triethylamine or inorganic bases like $\ce{K2CO3}$ are standard in this case and this is what they would have used if they had to do the reaction with a different amine (as the nucleophile) of higher cost.