I've had to use LRP instead of a Magnum primer many times in 300 WM and many receipts calling for a mag primers in 30 cal,25 and 243. I've seen no difference.
With due respect, if you haven't seen a change, you are not doing a thorough test. Changing the brand of primers, changing match and non-match primers, and most definitely changing large rifle to magnum rifle primers will cause a change.
Will all these different primers ignite the powder? Yes, most of the time ALL the powder will ignite, not always, but just because the cartridge went boom, doesn't mean you will "see no difference."
Using magnum primers will cause pressure to increase faster and usually require you to reduce your powder charge. Magnum primers will cause pressure signs to appear faster. The powder is burning faster because a magnum primer has more energy to ignite the powder column than a non-magnum primer.
A magnum primer will help ignition in extremely cold conditions, like -40F because it has more energy to ignite frozen powder. That's pure physics. It takes more energy to bring frozen water up to the boiling point than the energy it takes to bring luke warm water up to the boiling point.
Group size will be affected by changing primer types because all primers differ on the amount of energy they produce. Even the non-magnum primers within one manufacturer will change your group size, velocities, SD, ES, etc.
Can you get a magnum cartridge to fire using non-magnum primers? Yes, but are you using a high-speed camera and other instruments to see if all the powder is being burnt inside the barrel and thus contributing to velocity? Maybe you are ejecting a lot of unburned powder out the end of your barrel thus wasting powder and efficiency and increasing muzzle flash. Unburnt powder also contributes to faster barrel fouling, more cleaning, and faster barrel wear because the next bullet fired is grinding that unburnt powder into your barrel just like sandpaper.
Yes, primers make a difference.