John Tyler like most politicians of his day was not satisfied with the vice presidency.
Probably even more so than politicians of our day since in the 1830's, it was believed that in the event a vice president stepped into office after the death of the president, he'd merely act as the steward of the state until another election could be arranged.
We're probably much more concerned about who a presidential candidate is now than they would have been in Harrison and Tyler's time since we know that the VP will at the very least, finish out the term of the deceased president and potentially run for re election. Harrison supporters most likely wouldn't have had these Sarah Palin type concerns since they thought the worst that could happen was that the party would simply pick another candidate to run in a second election.
It's not as though he could have an alternative reading of the constitution, take the reins of power and refuse to step down could he? Well, yes he could apparently.
Immediately after news of Harrison's death came to him in his home in Virginia, Tyler returned to Washington and had himself sworn in. At this point, it wouldn't have been a clean process to unseat him from power, so people (most people anyway) grudgingly accepted.
This would not stop his enemies from addressing correspondences to him addressed to 'The Acting President' and people referring to him as 'His Accidency'.
It's ironic that Tyler set the precedent for a peaceful, orderly transfer of power in the United States since he took office against the will of almost his entire cabinet and leading members of his own party.