I think upgrading to LTE will make a substantial improvement. Remember that each train carriers multiple cell antennas for different carriers. Essentially, right now they get about 1.5 mb/sec from each carrier. With LTE, they will be able to pull in around 8-12 mb/sec per carrier. That's a huge increase, and if they need to, they can always add multiple cell antennas.
Sure, will coverage drop out in places with poor cell service? Of course. But those areas cover probably less than 3 or 5% of the total NEC, perhaps much less.
The maps shown in the article are also out of date. AT&T LTE, for instance, covers almost all of the NEC in Southwestern Connecticut (from New Haven to NYC). Verizon has committed to finishing the vast majority of their LTE buildout by the middle of this year.
I manage a WiFi system for a fleet of buses for a public transit agency. When we upgraded from 3G to LTE it made an absolute world of difference. While our buses seat a fraction of those on an Amtrak train, we also only have a single carrier per bus and we make no restrictions on high bandwidth usage, which Amtrak does a very good job of.
I am inclined to disagree with the Atlantic that things won't get substantially better. LTE represents huge bandwidth jumps and is available in more places than they suggest (keep in mind that carrier maps lag about 1-3 months versus actual deployed service area). Is shared WiFi ever perfect? No. But if it's that important to you to remain connected, bring your own cellular connection. Tethering is usually a $15 or $20 monthly option with most carriers.
And the cost to build out their own trackside network would be absurd and really unnecessary for the few places where cell service currently drops out. That money is much better spent on track improvements, rolling stock and the like.