HW is where the expansion really takes off in terms of story telling. However, if you skip the whole ARR, you won't really get as much of an impact that Heavensward gives.

They recently changed the storyline in ARR so the main story scenario quests are more streamlined. On one hand, they removed a lot of fluff from it (fetch quests included). On the other hand, some things that got removed actually helped build character developments and insight on the way they live if you like to learn about the world and the NPCs involved. Some dialogue was actually funny when playing through it for the first time (the scions and the Truesight Scalebombs for example, if you were to talk to each one of them during the story and during each part of the quest, they mention some funny tidbits about how they view their friends). However, because the quests were shortened and removed, some of that impact of the initial scene was really blunted. Subtle nuances about characters were removed in favor for a short description of the events that actually happened that doesn't really capture the whole picture and makes it feel like a quest just for the purpose of moving the storyline forward. As a player who replayed ARR 2.0, those cutscenes really didn't feel impactful as before but I at least understood the joke and implications because I remember what happened there. This is not to say the changes were bad, but it's just different. The impact is very different as the main character isn't as involved with the quests on a personal level anymore. You won't really notice the changes if you haven't interacted with most of the characters to to get a feel of their life as you progress through the MSQ. This is really apparent in ARR and HW iirc. Once you start Stormblood, a lot of the quest journal flavor text and NPC flavor text doesn't update as often as before or just get cut out entirely.

Does the game get better? - From a storyline standpoint and gameplay standpoint, yes since you have more access to skills and bosses start picking up speed at which mechanics fire at you. The story also stops feeling like you're a bystander and you feel more involved than before. Does the gameplay get faster? Your GCD is still 2.5 seconds, but you get skills you can use between GCDs so it does feel 'faster' but the overall speed is relatively the same.