In reaction to this and abuses by States, 14A was passed and ratified. It permanently took away the power of states to determine citizenship.
As for "subject to the jurisdiction thereof," it needs to be looked at for their time, not ours. There were no illegal aliens at the time, because there were no restrictions on immigration.
Jurisdiction: the right, power, or authority to administer justice by hearing and determining controversies.
Illegal aliens are subject to this in our countries, though it is sadly often unenforced. Legal armed invaders and those with diplomatic immunity are not.
By your definition, we have no right to prosecute or imprison those here illegally. They are not "subject to the jurisdiction thereof.”
Yes, it is true that everyone within the territory of the United states, is to some extent, “subject to the jurisdiction thereof;” otherwise illegals - those with no standing to be here in the first place - could commit all kinds of mayhem: murder, rape, arson, and be beyond the reach of our laws. But that isn’t what that clause was all about, and never was; it was meant to convey that the individual was “solely and uniquely subject to the jurisdiction thereof,” and not subject to the jurisdiction of any other country or political entity. Like Mexico, and if there might be any question about this, Mexico routinely issues their “matricula consulars” to their citizens in the United States, clearly indicating some level of jurisdiction over their own citizens abroad.
Admittedly courts have been all over the place on this language in the 14th, and there have been some very badly bungled cases, but I think that there is adequate legal scholarship to conclude that even at the time, the correct interpretation was as I have put it. That certainly was the position of the primary author of the 14th. I will have to dig out the reference for you.