The paradox given is an argument against perfectly efficient markets. The paradox is that if the market was efficient, then doing market research would be a loss-making activity always, meaning people wouldn't spend money to collect the information, thus making the market in-efficient.
Secondly, only a few percent of trading used to be done off-market. As of this year, more than 50% of all stock trades happen in dark pools, outside of the market, creating in-efficiency by definition, since the information is hidden from the market. The rise of passive investment also increases in-efficiency, since those buyers do not care how much the stock is worth, they simply buy the weighted basket.
Further if you are looking at funds, they are limited from being able to exploit in-efficiencies by design, since they are vulnerable to the main cause of inefficiency in the market which is the human psyche. For example if investors get scared and take money out, it forces them to liquidate at bad times, and if they put lump sums of money in, it forces them to capitalise even when it doesn't have good opportunities to trade on. Finally the size they operate at by definition distorts the market, thus meaning there is a limit on the size of bets they can make without reducing their alpha.
There are many traders with long-term track records that beat the S&P 500. Even Buffet, who famously recommends index funds for the average investor, has stated he doesn't believe strongly in the EMH, and obviously invests berkshires money in non-index funds, and over the past 60 years has beaten the S&P 500 handily, although it should be noticed as they have got bigger, they only beat it by a percentage point at best if at all, since any trade they make is so large that they distort the market heavily. Even so they show you can beat the market by going long. Also Charlie Monger his buisness partner seems to somewhat believe in a weak EMH.
In the medium term it is trivial to disprove the EMH, since we have now had COVID bubbles, the 2008 bubble, the dot-com bubble, the railway bubble etc.... All of which are practically impossible even if you only believe in the weak-EMH.
Finally quant firms pretty much prove that they aren't efficient on a short term basis either (though this is almost never disputed as you say!) since volatility existing is exactly this short term inefficiency and where they make a lot of money. Technically they are part of that restoring force that makes the market efficient theoretically.
A more accurate EMH would be something like "A stock is worth approximately what most people think it is worth at any one time". That is it is approximately efficient in regards to peoples perception of value at that time, as opposed to it's actual value. This is like the EMH rather tautological or even circular though, since this is literally the definition of the price of a stock.
In regards to me taking up day trading if I believe markets aren't efficient that is not the case. You can believe markets are not efficient without being able to find the in-correct prices yourself. You can also even believe markets are not efficient, and be able to find the in-correct prices and still struggle to make money because timing when the market corrects its in-efficiency is even harder, because the market can stay irrational longer than you can stay solvent.
And finally the classic joke:
Two economists are walking down the street when they pass a dollar bill on the ground. The first economist says “Hey there’s a dollar bill laying on the ground!” The second economist scoffs and says “No there isn’t, if there was, somebody would have already picked it up.”