Unless the residents you vote to murder are also given the same freedom to secede and draw a border that excludes you because they do not consider your rule legitimate, this certainly would not be democratic.
Your analogy illustrates why a right to secede matters: It is the ultimate peaceful (if the right is protected) tool for those who feel disenfranchised, whether by being robbed off the chance to vote or because they are a persistent minority, to ensure that either their rights are protected because the government do not want them to leave, or that they have an escape hatch against a majority abusing their power.
It is saying "we no longer accept that this government legitimately represents us," and the right and ability to do that seems to me to be the most fundamental concept of democracy - but only if it is extended to everyone, which takes away the problems of your example; you and your six friends would have no rights to draw a new border that includes other residents without they too having a say, including the right to themselves secede if you do not offer them something they are willing to see as legitimate.
As such it provides an important incentive against over-reach and towards negotiated settlements that does not exist when there is no realistic mechanism for a minority to vote for their region to leave.
You can mitigate the need for a right to secede by giving sufficient protections for minorities against the choices of the majority; but whenever secession gains substantial support, that is evidence that whatever mechanisms are in place are insufficient.
To me, a government that feels a need to deploy police to stop a region from demanding independence is inherently illegitimate.
At the same time, I am all four having secessionists be made to understand the consequences, in that if you a house and decides your house should secede, then fine, but you e.g. have no inherent right to then be allowed to cross the border, or expect your new neighbouring state to provide you with any services, and you can expect them to act with force to protect the interests of any of their citizens resident on "your territory" who do not want to secede and who are not offered sufficient protections - including their own right to secede from "your state" and rejoin their preferred state.
In practice I think that taking such a right to secede to it's full consequence would minimize actual uses - anyone wanting to do it would need to take into account the problems of whether or not they'd e.g. end up with enclaves, eroded borders, exclaves, and whether they'd even end up with a contiguous territory of significance at all if they have majority support in a region but also lot of resistance, and would be forced to actually negotiate to solve such issues in a way that is not inherently detrimental to both sides. At the same time this would also apply to the state you seek to leave.