I guess there's no benefit to pursuing "independence" like that. IMO the question is like asking, "Why doesn't Bill Gates mow his own grass? What would happen, if he lost access to a gardener?"
I gather from the news, for example, that what keeps Iran from having military nuclear weapons is centrifuges -- i.e. that you need a lot of good centrifuges to separate fissile Uranium. IMO a country like Germany for example isn't officially 'a nuclear power', but looking at its industries I think it would obviously be able to manufacture any tools like that (i.e. like centrifuges) that it might ever need.
Your mentioning "Facebook" implies to me that you're picking the (few?) areas in which 'American' companies dominate, and asking, "Why doesn't Europe compete?" I presume that Facebook is trivial technology: merely a web site, a famous name, and users. Everyone these days (i.e. Europeans too) knows how to write "web-scale" sites. The leading cloud providers (Amazon, Google, Microsoft) are OK or leading-edge at what they do, but IMO what they do is just (or is mostly only) a matter of money -- i.e. anyone could do that, if they had to and were willing to pay more, to do it more expensively than buying off-the-shelf from the incumbents (e.g. governments presumably do run their own relatively expensive, private, secure, computer networks and servers).
Also Google for example has development offices in Europe -- in Switzerland, and so on. My guess is that if (hypothetically) America ever went dark, off the grid, the fundamental technology or know-how is well distributed globally. If it isn't commercially viable to compete against Google -- and they have a lot of trade secrets, and own a lot of assets (hardware, intellectual property, and human resources) -- but even so IMO it would be technologically feasible to deploy a inferior national-scale replacement implementation in the event of a national emergency.
IMO there's limited national advantage to "owning" something like Facebook (one could argue that it obviously didn't do America any good). Web and software companies (like Amazon or Uber) have a bit of first-mover advantage, and spend a lot of money (i.e. are unprofitable) to get there -- usually private money, rather than government subsidies and investments -- but even so have local competition which could take over if the 'American' company disappeared.
The internet itself is distributed --- a country can be cut off from others and still have its own web sites: which is maybe more important for countries like China (has a firewall and censorship, and ambitions of its own) and Russia (may face sanctions).
If the governments (e.g. EU) are going to spend money on subsidising any industry, there are other industries that are more "strategic" than you ones you mentioned -- e.g. the national power grid; public transportation (including trains) and roads; agriculture (including "food security"); national health (medical research and deployment); national education; the telecommunications grid; the tax office and the police force; etc.
I'd argue too that laws and regulations (political science) are an important technology, and are the proper domain (sphere of interest and influence) of national governments, and something which the EU does pursue "independently".
And Europe does invest some in far-out R&D -- outer space; nuclear fusion; CERN; not to mention military (and medical and industrial) devices.
And countries do (to a certain extent) maintain some independent social media (propaganda outlets) too -- e.g. the newspapers, TV stations, radio broadcasters aren't all foreign, they may have some indigenous movie production too, and any number of non-English-language web sites and/or government web sites that you won't have heard of.
Do EU countries actually have most of these technologies, even if in a latent form, not visible to consumers? Or perhaps they do try to achieve technological independence but simply fail at it?
I think they have technological ability.
And that what they don't have is the political will to spend national money, to subsidise "private industry", to try to ensure that their "national" companies become world-leaders in every industry.
Some of "their" "national" companies are (already, naturally) world leaders in their industries (Rolls Royce for jet engines).
I guess they have neither the (financial) capacity nor the political will required to subsidise (or inflate) an ambition to be "world-class" (or world leaders) in every industry.
That's more-or-less true of every country, I think -- every country has some industries or market niches in which it's out-competed in the world market.