Well it sounds like you have 2 problems.
#1 You need to replace the sump pump. That pump was probably put there for a reason or someone had too much time and money so they got board and put one there. So yes why not replace it.
#2 Sounds like you had a blockage in the toilet line for the powder room. It sounds like a vent line is blocked by the given description. But if the plumber says it was a sewer line being clogged that could of lead to the vent getting clogged. Depending on the layout of the house (plumbing) a 50 foot snake down the roof could fix the vent.
Questions for you…
Do you have a septic system?
Dose the sump pump plumbing go right into the household plumbing or can you tell?