The execution doesn't quite live up to it.
After The Daleks and The Dalek Invasion of Earth two great Who serials and obvious evidence as to why the Daleks became such a huge cultural phenomenon, The Chase is their first real let down of a story.
It just never lands well. The bulk of the middle feels like the story is just spinning its wheels. The third and fourth episodes do what you expect from a serial called The Chase. The Doctor and his companions trying to get away. But it's only three stops, and they drag on far too long. The boat works, but the top of the Empire State Building and the House of Horrors just dragged. New York mostly because it decides to have a comedy scene with a yokel instead of anyone panicking about Daleks. The revelation that Ian and Barbara are more or less in their home timezone, if on the wrong side of the Atlantic is completely ignored too.
Frankenstein's monster and Dracula just goes on far too long, that little escapade takes an entire episode. It doesn't move the plot on in the slightest, except when they leave Vicki doesn't make it to the TARDIS door and has to stowaway on the Dalek's time machine. That the Doctor decides that somehow they'd entered the realm of human conciousness and not just a house of horrors was a great touch.
"The Death of Doctor Who" episode is just a terrible fake-out. It's blatant from before the episode even starts how they're going to live up to that title without actually killing off their star, and the show does nothing with actually turning that concept into anything resembling tension. The audience is never once confused about which Doctor is the real one, though the companions are. The only good part is the reveal of how Barbara realises when the robot Doctor calls Vicki “Susan” because the Daleks don't realise Susan left the TARDIS and Vicki took her place.
Nothing in the preceding 5 parts has anything to do with defeating the Daleks. There's that strange box the Doctor and Ian were carrying around, but that kills all of one Dalek. Everything else is the Mechanoids, who are so badly designed it's laughable. Even the TARDIS crew's escape is down to Stephen, who again is introduced in part 6. The only thing that plays any significance from the previous parts is the Daleks' time machine, which promptly gets Ian and Barbara home.
The Doctor does nothing to bring resolution to the whole story. The only thing that's stopping the Daleks from chasing them across all of time and space again is that Ian and Barbara nicked their time machine, joyrode it home, then blew it up. As soon as they build a new one, The Chase should begin anew. It's a shame then that the next Dalek serial The Daleks' Masterplan is missing. I'm not sure if it even deals with it, but The Chase is so poorly resolved I need something.
However, the main thing of The Chase isn't that it's another Dalek serial. It's that in the final episode Barbara and Ian decide to leave. The second half of the last episode is fantastic, and about the only good thing going for the whole serial. Even the fact that they actually made it home to 1966 halfway through but just got back in the TARDIS as they stopped in New York can be forgiven.
Our intrepid teachers realise what they have and announce they intend to take the Dalek time machine home, and the Doctor loses it. How vehemently he fights against their leaving, and how distraught he is when he finally comes round. It sets the tone for how he always sees companions and when it comes time for them to move on. It's quite the contrast to how they joined, when he couldn't wait to get rid of them.
This isn't the farewell he gave Susan, the family member he practically pushed out of the door so she could live a proper life, but to also remove some of his responsibilities.
We're back to looking at just how much of an effect Ian and Barbara had on the Doctor. The show starts with him already in exile from Gallifrey (though that won't be established for another few years), having already had a few limited adventures. But as those two come on-board he embraces his adventurous spirit, and with the departure of his grand-daughter he finally becomes the Doctor that has endured fifty years plus. Now they've gone. That's a pretty momentous moment. Yes, Ian was meant to come back years later, but thanks to a schedule conflict his role was fulfilled by the Brigadier in Mawdryn Undead. It is these two companions, the original two companions, that are wholly responsible for the Doctor becoming the man we all know. That's pretty important.
Not only that but William Hartnell is left as the only original cast member. Everyone talks about Doctor Who's ability to reinvent itself as a programme when the Doctor changes. I once read a comment (probably at TARDIS Eruditorum) that fans of most other TV shows moan when their programme is cancelled. Yet Doctor Who fans have to put up with it every three or so years. But that's true of the companions as well. At least when you have such a seismic change as here. Susan to Vicki just allowed the young girl to be bit pluckier and less screamy.
Removing Barbara and Ian, fundamentally changes all the relationships within the show. We may have the same Doctor, but everything else is different. Imagine Friends if the boys had moved out and the show continued with someone else living across the hall to Monica and Rachel.
Looking more recently, the 11th Doctor travelling with Amy and Rory was very different to him with Clara. Not to mention Clara and 12. We started with a show that was based around a character, his granddaughter and her two teachers. The granddaughter left, and now her teachers follow. The show becomes the Doctor and a young female companion. What became the default.
Or does it? The Chase leaves the fate of Stephen Taylor totally and utterly open. As far as that episode is concerned he's left wandering the lethal forests of Mechanus, it's only because I know what's coming up that I know Peter Purves becomes a regular.