No, we don't have "free trade", but not because of contibutions paid to be part of the EU. You are quite correct that those contributions have nothing to do with the "free" part of free trade.
However, the EU is not a free trade area. It is a single market, though very much an incomplete one.
We have partial free trade, but not wholly so, within the EU, but even on goods it's not entirely complete and on services, which by definition should normally be included in a free trade area, which were largely included in the original pre-single market EU, and certainly have been repeatedly promised by the EU, are woefully short of happening.
In theory, the EU is a singie market, which takes the usual free-trade principles of removing taxes, tariffs, quotas etc, way further by also completing a "level playing field" by subjecting all participating members to exactly the same operating constraints, like health and safety standards and rules, packaging and labelling regulations, chemical content standards and much, much more.
So, in some significant areas, like services, the EU falls way short of even free trade let alone single market requirements, but in others, like free movement of capital, people, etc, it goes way beyond free trade.
The truth is, even if we left the EU and the singke market, any UK firms seeking to export to the EU would have to comply with many of those requirements, regulations (like labelling, etc) but only for goods they sought to export to the EU. Companies here not seeking to export to the EU would not have to comply, while currently, as part of that incomplete EU, they do.
Of course, the rest of the planet seeking to export to the EU manages to produce goods that comply with the EU regulations without being part of the single market. The US, China, entire south Pacific, south American regions, Canada, India etc all trade. And any EU member seeking to export into any of those countries has to comply with regulations, etc, of the country they're exporting to. If I want to export to Australia, I have to comply with all relevant Aussie regulations to do so, on goods I wish to export there.
Ae do not have free trade with the EU. We have mostly free trade, gold-plated with single-market add-ons, for most goods and far less services. And as the services sector is critically important to the UK, the degree of free trade we have we the EU is far less than you might think.
Moreover, the whole argument is missing the point, which is that the issue is far more complex than that. For instance, single market membership imposes a substantial regulatory burden on firms that either to not export, or do not export to the EU, with no compensating benefit from that single market. Such firms have to comply with such EU impositions, then if they want to export to Australia, the US etc, they have to comply with national requirements there, too, even though they are different to the EU.
Nobody can say for certain what the impact of leaving the EU would be. But when being part of a single market also implies being part of a bloc that imposes extra burdens to countries outside it trading in, as well as inside it trading out, then we not only have to consider the benefits of the 500m people in it, but the burdens imposed on the 6.5bn not in it, and the trajectory of those economies and our trade with them.
What worries me, in purely trade terms, is that being part of the EU is rather like hand-cuffing ourselves to the ship to ensure we don't get washed overboard, blissfully ignorant of the fate of our ship, the Titanic. Any half-competent economist can't fail to be aware that the EU ship is sailing full-speed through a minefield of icebergs, and given the state of many economies including some big ones, is already holed below the waterline, is taking on water and there's a huge questionmark over whether the bilge pumps can cope.
Oh, and a hell of a lot of the passengers are threatening mutiny and would live to lynch the captain and senior officers.
Should we be hand-cuffing ourselves to that, or booking our place on the life-raft?
I haven't yet finally decided, but I've geen waiting for some argument from Remain to convince me and have so far been disgusted by the nature of the campaign. Not that Leave is much better.
So, I'm voting out in this thread, but am still open to convincing otherwise if the Remain csmpaign can get their heads out of their butts and make a rational, not hugely distorted, case.
In the end, the EU is not about trade. It is and always has been a political project designed to produce, whatever we call it, a unified European state to counterbalance the Russians, the US, China, etc, and what we're really asking ourselves is not about short-term trade but about whether we want to be a small (but still economically significant) country, or an important region in a single European state?