Instead the Sun is predicted to expand as it "dies", as other stars of similar size have done. That could spread its extremely hot atmosphere out as far as the Earth or beyond. So even if the planet itself survives all life on it will be destroyed.
"One day" though is some thousands of millions of years from now.
The model you ask, that of the Sun and Earth approaching, would be the other way round. The Sun would not approach its planets but they might approach it. I don't if that could happen or if instead the planets would slowly spiral outwards, away from the Sun.
I think it will take another billion years for the Sun to grow so large, and make the Earth so hot that all life will cease to exist. Then it's Mars' turn to become what the Earth once was.
My machine can't pick up anything directly from Tic Tok - ty for making the vid available. I feel like a dope - LOL - I was expecting the elderly actor Brian Cox...
In a few billion years the sun will become a red giant as it uses up the last of its fuel. It will expand massively and as it does the earth elbe swallowed up or at least torn apart by the gravity of the sun before the sun dies and collapses to become a white dwarf.
In approximately 5 billion years, the Sun will evolve into a red giant. This transition will occur after the Sun exhausts the hydrogen fuel in its core, causing it to expand and cool. The Sun will eventually become large enough to engulf Mercury, Venus, and potentially Earth, before collapsing into a white dwarf.