Does introduction of new variable always increase the p-val of existing ones? I am doing some work that requires some estimates of gasoline oil demand elasticity on certain countries. After doing various econometric measures such as instrumental variable, I was able to get reasonable estimate of demand elasticity for most countries, but only for some special ones (mostly in Europe), the values are confusing (positive for example) and the p-val is huge. Is it safe to assume that for these countries the price is just insignificant to Demand? Specifically, I am wondering if I add any other variables, in practice would it only increase the p-val of price and make it less significant?
Say the model I currently have is $$\log Demand = \beta_1 X + \beta_2 \log Price + \epsilon,$$ whereas the new model is $$\log Demand = \gamma_1 X + \gamma_2 \log Price + \gamma_3 Y + u.$$
Further suppose that $Cov(X, log Price) = 0$ (not true), then I think we have the equation that $$\gamma_2 =   \beta_2 - \gamma_3\sum_i \frac{Cov(\log Price, Y_i)}{Var(\log Price)}.$$
So if the new variables are correlated to $Price$ then the multicolliearity will just increase their p-val. If they are not correlated, then $\beta_2 = \lambda_2$, and you would theoretically wind up with the same estimates, with a higher SE because of the increase in dgree of freedom, and therefore get a higher P-val. (p.s. I know this argument is not sound in theory, but in practice does it make sense intuitionally?)

Definitely not, and there is a sense in which this is why we do regression.
An example is analysis of covariance (ANCOVA), which is the usual ANOVA but with an additional variable that typically is called a covariate.
I’ll give an example with two groups.
set.seed(2022)
N <- 100
g <- rep(c(0, 1), c(N, N)) # Group
x <- rep(seq(0, 1, 1/(N - 1)), 2) # Covariate
y <- x + g + rnorm(N, 0, 3)
L1 <- lm(y ~ g)
L2 <- lm(y ~ g + x)         
summary(L1)
summary(L2)

In the first regression, I get a p-value of about $0.025$ for the group variable, while the second regression gives a p-value of about $0.018$ for the group variable.
The reason that I say this is why we do regression is because this is helping us increase the signal-to-noise ratio, thus allowing patterns (signal) to become apparent. If your threshold for evidence of a difference is a p-value under $0.02$, then the first regression gives insufficient evidence that there is a difference between the groups. However, once we use that covariate to explain some of the variability (noise), the pattern jumps out from the background.
EDIT
While this was not the original question, for completeness, I will add that adding a variable does not have to lower the p-value. One way for this to happen is if the new variable is correlated with an existing variable.
library(MASS)
set.seed(2022)
N <- 100
X <- MASS::mvrnorm(N, c(0, 0), matrix(c(1, 0.9, 0.9, 1), 2, 2))
x1 <- X[, 1]
x2 <- X[, 2]
y <- x1 + x2 + rnorm(N, 0, 3)
L1 <- lm(y ~ x1)
L2 <- lm(y ~ x1 + x2)  
summary(L1)
summary(L2)

This goes from a tiny p-value on the order of $10^{-12}$ to a p-value above our usual threshold of $0.05$.
