Why can't I just do a "normal" regression to get the coefficients?

Because it is not numerically stable. Remember computer is using fixed number of bits to represent a float number. Check IEEE754 for details, you may surprised that even simple number $0.4$, computer need to store it as $0.4000000059604644775390625$. Using raw polynomial will case a problems if the number is very large.

You can check my answer here for an example.

Why are there large coefficents for higher-order polynomial

Why can't I just do a "normal" regression to get the coefficients?

Because it is not numerically stable. Remember computer is using fixed number of bits to represent a float number. Check IEEE754 for details, you may surprised that even simple number $0.4$, computer need to store it as $0.4000000059604644775390625$. You can try other numbers here

Using raw polynomial will cause problem because we will have huge number. Here is a small proof: we are comparing matrix condition number with raw and orthogonal polynomial.

> kappa(model.matrix(mpg~poly(wt,10),mtcars))
[1] 5.575962
> kappa(model.matrix(mpg~poly(wt,10, raw = T),mtcars))
[1] 2.119183e+13

You can also check my answer here for an example.

Why are there large coefficents for higher-order polynomial