Are Lagrange Multipliers necessary for Shamir's secret sharing? What is the implication of not using the Lagrange multipliers in Shamir's secret sharing?

I mean each update will be $Poly(x_i) \bmod p$ where $p$ is a prime number instead of the original one that is $(Poly(x_i)* LPC_i) \bmod p$. 
Assuming that the verifier has access to all secrets $x_i$s, it can re-do all the updates and check if the result is equal to the secret or not.

Is that less secure?
I will answer you by an example, assuming Shamir sharing algorithm with $d=2$ (minimum shares). So, by definition we need a $d-1$ degree polynomial as:

$Poly(x)$ = $ A_1 \cdot X^1 + S$ = $10x +20 $ where $S=20$ is the secret. The dealer will calculate the shares as (two shares for simplicity, where 2 is needed to reconstruct the polynomial):

$(1) [x_1, Poly(x_1) ]$ = $[ 5, Poly(5)]$ =$[5,70]$ , and 

$(2) [x_2, Poly(x_2) ]$ = $[ 10, Poly(10)]$ =$[10,120]$.

Now, without using Lagrange interpolation: the verifier needs to reconstruct the polynomial equation (i.e., line equation) to find $S$ as:

$(1) Poly(5)=A_1 \cdot 5 +S =70$ and 

$(2) Poly(10)= A_1 \cdot 10 + S =120$

So, you will need to solve the two equations. This is a simple one because it has only two equations. However, for larger  $d$ minimum shares it will be intricate process to solve lets say 10 equations.

Second, by using Lagrange interpolation it will be much easier because we are only interested by $Poly(0) \equiv S $. 

In conclusion, both have the same security but Lagrange interpolation is much faster way.