I think you've all correctly diagnosed the problem. New England is hugely dependent on natural gas for its electricity and for home heating. The two major interstate pipelines from the west max out their capacity when it's cold (heating demand is high).
You might reasonably ask why the pipelines weren't built big enough to meet the needs of retail gas customers and the gas-fired power plants. The answer is simple but ultimately is an unintended side effect of electric restructuring ("deregulation"). To build a new gas pipeline, you need FERC's permission. FERC only grants that permission if you have contracts for nearly all of the capacity you plan to build. Gas distribution companies, those that serve retail customers, are willing and able to sign such contracts when they see a future need, because they can recover the associated costs through regulated rates. What about power plants? Every large gas-fired plant in New England is owned by a non-regulated company, so they must recover all their costs through the sale of electricity and related products. But you don't earn a premium for having firm gas delivery rights, and if you were to buy such rights and found that you weren't dispatched fully, you'd have to resell those rights (probably at a loss). So, there's no financial incentive to contract with pipelines; indeed, quite the opposite. Even though we know collectively how much gas we're likely to use as a region, no individual market participant knows what fraction of that total will be used at its plants. Without some central planning entity on the gas side, we just won't build as much pipeline as the region can be expected to need.
You might ask, why doesn't some investment bank or other third-party just buy the transmission rights, knowing that there's a price spread? The trouble is, as soon as you build the incremental capacity, the price spread collapses. Buyers of the transmission who don't have a financially firm contract to monetize today's price spread will lose.
All this comes at a huge cost to the consumers. The cost of the price spread between New England delivery and Pennsylvania delivery, times the cost of all the natural gas used in New England, is multiples above the annualized cost of building a pipeline that would eliminate that price spread.