How can TINA possibly explain the U.S. stock market rally in 2020, when U.S. interest rates before Jan 2020 was < 2.3%?

If TINA is true, wouldn't investors have reallocated to buying stocks way earlier than Apr 1 2020?

They did. If you look at the long time frame, the stock market had an unprecedented bull market for a decade, fueled to some extend by low interest rates.
Even in 2019 bonds where not really an attractive investment. Germany for example had already placed bonds with negative interest rates as far back as 2012. This is not negative after inflation rate, this is literally negative. While the US had slightly better conditions, this is going into the same direction. So, while your quotes suggest TINA is explaining the 2020 rallye it has already been there long before.

But TINA is actually a part of the rallye. In early 2020 people were pulling out their money from the stock market but what do you with the cash? It has to go somewhere and there simply was no alternative. So people started investing into the tech sector. After all, there is no better stimulus to Amazon than closing down shopping malls. As the tech stocks are quite heavy weight, they were pulling up the S&P 500. But this is only half of the truth. Many sectors were left behind (oil, tourism, industry) and are only now catching up as the end of lockdown measures is near.

I'd offer that over $3 trillion of stimulus money pouring into the economy and the stock market was the driving force of market gains:

Investors poured record $56.8 billion into stock-market funds as stimulus checks arrived.

Experts concluded that the initial CARES Act boosted the market considerably. Many people invested part of their stimulus checks in the stock market. Schwab saw a record number of new accounts opened and Robinhood saw daily trades increase 300% in March 2020, year-over-year.

If TINA is true, wouldn't investors have reallocated to buying stocks way earlier than Apr 1 2020?

They sure did and that's the reason why the stock market is so bullish and dangerously overvalued. A big driver behind this are retirement savings (which are mostly done on auto pilot)

As Social Security is being faded out (and/or gets eaten by rising health care cost) the next best viable options are taxed deferred retirement savings plans (401k, IRA, Roth, etc). Despite economists and the media constantly bemoaning the poor savings behavior of Americans, the total yearly contributions to IRAs alone is around a quarter of a trillion dollars each year.

Here is where TINA comes into play: all this money needs go somewhere. Bonds & treasury have negative real returns so stocks seem the only option with some growth potential.