Moreover, it has 0 influence on the way you wage war.
In the MP game we just completed, infrastructure had a huge impact on how we waged the war. There were at least three theaters where either damage to infrastructure or just poor starting infrastructure regulated the war to a large degree. Despite facing a determined opponent in the form of Germany and the Luftwaffe, Britain (me) had to keep spending CIC on infrastructure upgrades in a couple of crucial areas so we could maintain the fight in Sudan. I'd have much rather spent more CIC on more factories, but we needed the infrastructure.
My last MP game as the Soviets, infrastructure played a crucial part in two different counter-offensives against the Germans. Not only that, but in almost every game as the Soviets, whether I win or lose, damage to infrastructure (and who has the right to repair it in any particular state) regulates a substantial part of the logistics side of the war. I can't count the number of times I'm looking at the logistics screen trying to figure out a way to eek out another 1-2 supply in a region so I can squeeze in another division or wing of planes. Upgrading infrastructure (in the areas the logistics map tells me are bottlenecks) is a part of that.
I'd also like to note that infrastructure level dictates the
speed at which you strategically redeploy. Doing it in a high infrastructure state is faster than doing it in a low infrastructure state.
I won't say that the system couldn't use improvement (in our MP mod, we changed some stuff to make attrition harsher across the board and make certain weather/climate much harsher), but I will say this: supply grace masks a lot of the depth in the supply system and just how important infrastructure is.
When facing the AI or a substantially inferior opponent, supply grace lets you do amazing things. You can make an initial breakthrough, encircle 1.5 million Red Army troops, and wipe them before your supply kills your offensive. You can make an initial breakthrough and seize VPs (which generate local supply) to keep the momentum of your offensive going. When facing a competent opponent, this kind of thing is much harder to pull off. And if you can't complete massive offensives before supply grace runs out, suddenly infrastructure matters a whole lot more.
Hell, we've been talking about nerfing the FM supply ability in our MP mod due to how much impact it can have in certain cases. That's how important supply, infrastructure, and logistics can be.
I will also say something else: units in HOI4 tend to move pretty damn fast regardless of how you feel about strategic redeployment. We ended up just slowing everything down in terms of division movement, and it made for what we consider better gameplay. (We also nerfed the impact from air superiority as well to prevent stunlocking.)