Can L1 Lasso regularization can produce L2 Ridge regularization results? Today I learned that the L1 Lasso's lambda parameter can be adjusted from small to large, with small being that Lasso basically produces the same result as least squares (no regularization/ leaves coefficients as is), and large being it shrinks all coefficients towards 0.
And so the practical approach would be having that lambda value somewhere in between, likely shrinking unimportant coeffs to 0 and keeping important ones above 0.
My question is isn't there a lambda value somewhere in that spectrum that would represent the same output a Ridge regression would output? I'm imagining this lambda value slightly smaller than shrinking any coeffs to 0, as per the definition of Ridge.
If this is possible, why do we even bother with Ridge if we can just adjust the lambda value accordingly to our needs (depending on if we wanted a Lasso or Ridge output)?
 A: Here is an simple illustration in R of why this is unlikely to happen when you have more than one explanatory variable and they do not have equal coefficients, taking an example where by construction $$Y = 2X_1+X_2+\epsilon$$ with $X_1,X_2,\epsilon$ iid $N(0,1)$
set.seed(2023)
nobs <- 10^5
X <- matrix(rnorm(2*nobs), nrow=nobs)
Y <- 2*X[,1] + X[,2] + rnorm(nobs)  
summary(lm(Y ~ X))
#Coefficients:
#(Intercept)           X1           X2  
#   0.003684     1.997131     0.996772  

coeffs <- matrix(nrow=201,ncol=5)
colnames(coeffs) <- c("lambda", "lasso1", "lasso2", "ridge1", "ridge2")
for (n in (-100):100){
  lasso <- glmnet(X, Y, lambda=1.1^n, family="gaussian", alpha=1) 
  ridge <- glmnet(X, Y, lambda=1.1^n, family="gaussian", alpha=0)
  coeffs[n+101, ] <- c(1.1^n, lasso$beta[,1], ridge$beta[,1])
  }   
plot( coeffs[,"lasso1"], coeffs[,"lasso2"], col="red",  type="l")
lines(coeffs[,"ridge1"], coeffs[,"ridge2"], col="blue", type="l")

Plotting the coefficients of the LASSO regression in red against each other and similarly the ridge regression coefficients in blue, your assertion would require the two lines to cross somewhere.
They do not in this example except at the extremes of zero shrinkage and total shrinkage (and would not in most other examples) because the LASSO regression model tends to reduce them by similar absolute amounts until one disappears, while the ridge regression tends to reduce both by similar proportions with both falling towards $0$ together.

