Thiel is correct. Any centralized system wherein multiple cultures must fight for a majority with which to impose their culture on everyone else, does not respect the freedom of slightly under half its citizens.
("Red Tribe" and "Blue Tribe", as one may call the collection of traits held by the average conservative and average neoliberal, respectively, are effectively different cultures[1].)
On the other hand I'm not sure how this essay's viewpoint would help in this regard. From my vantage point, Thiel doesn't seem to have much respect for the opinion of other voters who don't share his philosophy, with two groups ("women" and "welfare recipients") called out specifically. It seems like there is little attempt to figure out why not every person is enthralled with Thiel's brand of libertarianism, and much of the essay is devoted to dreams of tribal utopian colonies (which historically often end in disaster). The impression I get is that he views himself as better than almost everyone else, to be honest... that's not a very likable trait if so...
Oddly enough, I personally find that Thiel's support and stances in relation to Gawker vs. Hogan a bit contradictory to this dream. This is less a commentary on the lawsuit (I found Gawker detestable personally) but more how Thiel's Gawker vs. Hogan lawsuit embrace is a dip into the very same complex American legal system that usually is harshly criticized by libertarians.
You can enact laws that respect and support multiple points of view. See: national service but with exemptions for reasons of faith, mixed public private medical systems etc.
Also, I would say that you are using a shallow definition of culture.
As best I understand, having a single system with aspects of both extremes, has not brought out the advantages of either. We'd be better served by having a public system and a market that competes with it; except that nobody in good health would find the public system to be competitive, and that thus the public system would devolve into a public charity for the chronically ill/injured/etc.
> national service but with exemptions
That's all well and good, but exemptions become more difficult for a number of other things.
For instance, rural residents (a.k.a. "Red Tribe") generally prefer to be taxed less and receive less subsidized services overall in return; whereas many urban residents wish for a sort of Northern-European "nanny state" which provides every service imaginable in exchange for having negligible post-tax net income.
And here's another example which speaks more directly to geography. In rural areas there are (a) hostile and/or food-bearing wildlife, (b) large wooded areas devoid of humans (or, at least, demarcated with warnings that all humans within must wear high-visibility vests) and (c) an abundance of soft, bullet-absorbing ground. For this reason, rurals see firearms as a useful tool that can be handled safely enough to not cause injury to humans or damage to human property. (Unless, of course, one is an outlaw who intends to do so.) In urban areas, by contrast, there is scarcely any direction at all in which one can point a muzzle without "flagging" something valuable or someone; either directly, or on the ricochet from the hard materials that are common in urban areas, or even penetrating through a wooden wall/floor/ceiling of your apartment. It is no surprise, then, that many urbans see no purpose to civilian firearms ownership whatsoever; and would never see such a thing without entering a rural area; and would thus have no empathy or respect whatsoever for their rural neighbors who are so wary of firearms restrictions.
("Red Tribe" and "Blue Tribe", as one may call the collection of traits held by the average conservative and average neoliberal, respectively, are effectively different cultures[1].)
[1]: http://slatestarcodex.com/2014/09/30/i-can-tolerate-anything...