The real game changer for battery trains is the introduction of solid state batteries.
www.ufinebattery.com did a great article comparing the difference between them and the standard lithium ion ones used today but in language that doesn't require an engineering degree. Solid state are superior by every metric. Compared to lithium, they are 9X safer, 2.3X more energy intensive, recharge 3X faster, last 1.7X longer, and weigh 30% less while being smaller. They, unlike lithium, can handle any extreme temperature condition.
Battery and catenary trains are exactly the same and completely interchangeable offering much smoother, quieter, emissions-free travel while not having the diesel fuel expense. There is however, one very clear advantage catenary has over battery and that is acceleration. Battery trains often don't have near the acceleration and usually have top-speed of the same train about 20% less. They are somewhat faster than diesel but not markedly so. This has nothing to do with the engines or train design and everything to do with the monstrous dead weight of carrying around the batteries.
Solid state batteries offer a quantum leap in negating the BEMU drawback of battery weight and hence performance. By weighing 30% less, requiring less than half the number of batteries for the same amount of power generation and even fewer batteries on top of that because they can recharge so fast, battery weight could easily be reduced by 75 to 85% depending on things like distance and station stopping. Still not catenary speed because it still represents some dead weight but the difference in performance would shrink very significantly and could truly be considered rapid transit. The performance difference will continue to shrink as batteries and regenerative breaking technology continue to advance. Being vastly safer, means usage in tunnels or enclosed areas {ie Union} is not an issue.
The problem is that solid state batteries currently cost about 8X more than lithium due to lack of scale but that is changing at a monumental rate. Already mass production is starting this year and in just 4 years, price parity is expected. Interestingly, the new VW gigafactory in St.Thomas will begin mass solid state battery production by about 2029.