First of all, why not both? Why is one (easing domestic production hurdles) tied to the other (restricting export) in your argument?
But I'll take a stab. More of something means cheaper prices assuming demand remains the same. And oil/gas is a global market regardless of our export, or lack of export, restrictions. And demand worldwide is highly inelastic. The price goes up and people just pay it and if its a business or production input the increase just gets passed on down. So if we produce more oil and gas and everyone else stays the same, it gets cheaper eventually. Smarter people than me have posted reasons why you can't index the price at the pump to the price of oil in this thread.
If the gloves were off for US producers in terms of no more restricting federal leases, no slow playing permits and no ESG bullshit choking financing and investment and we embraced the fact that a barrel of oil not produced here is a barrel that will be produced elsewhere, if possible, then we would see prices fall over time. Nothing anyone can do is going to reduce the price people pay at the pump over night.
As for pipelines...they are objectively safer and more efficient than literally every other means of transport. So, do with that what you will if you are wondering why Keystone XL got torpedoed.