Before answering the question of what is the point of saying that there is no general formula in radicals for the roots of a quintic I would pose a preliminary question:
What is the point in having any formula for the roots of any polynomial?
If the goal is to compute numeric approximations for the roots of a polynomial with known numeric coefficients then you do not need any formula for the roots at all. Instead square-free factorisation and numerical root-finding algorithms can compute the roots of polynomials of any degree more accurately than any general formula. There might be certain situations where it is more efficient to use something like the quadratic formula rather than a root-finding algorithm but certainly no formulae are necessary to be able to compute the roots numerically.
In practice the value of having explicit formulae for the roots of a polynomial comes from using them as part of a symbolic calculation where the coefficients of the polynomial are symbolic expressions rather than explicit numbers. For example the differential equation
$$
m\ddot{x} + b\dot{x} + kx = 0
$$
could describe a mass on a spring using symbols $m$, $b$ and $k$ as the coefficients. Explicit expressions for the solutions to the differential equation using radicals can be found using the quadratic formula and the behaviour of the solutions can be analysed by looking at how they depend on these symbolic parameters. Since this differential equation is 2nd order we would want the quadratic formula but for 5th order we would need the roots of a quintic and for a system of $n$ coupled masses we would need the roots of a polynomial of degree $2n$. This differential equation is just one of many examples where a symbolic calculation might lead to wanting symbolic expressions for the roots of a polynomial and where increasingly complex problems lead to polynomials of higher degree.
In this context we can use the quadratic, cubic or quartic formulae for polynomials up to degree 4 and obtain expressions involving radicals. Further symbolic manipulation of these expressions involving radicals is possible but can be awkward. One difficulty is that radical functions are generally awkward to work with and another is that e.g. the quartic formula gives large complicated expressions for the roots. Galois theory and the Abel-Ruffini theorem tell us that for degree 5 or more we might not even be able to get a solution in radicals and would instead need to work with more complicated functions that would be significantly harder to manipulate in further symbolic calculations. This problem is compounded by the fact that non-radical formulae that do exist for unsolvable quintics are also significantly more complicated than the quartic formula which is already unwieldy enough to be of questionable usefulness.
In practice even for symbolic calculations it is usually better to avoid using radical formulae whenever possible. There are however classes of problems that can be handled symbolically via the use of explicit formulae although doing so becomes more difficult for higher degree polynomials. The Abel-Ruffini theorem represents one hard limit on the approaches that can be used when attempting to scale up from simple symbolic calculations using the quadratic formula to more complicated problems involving higher degree polynomials.
