I think John 5:24 means that anyone who believes in Him has the life.
But I also believe and take care that my heart doesn't become evil and unbelieving and hardened as with Israel in the desert. Who was God displeased with and swore that they would not enter His rest? Wasn't it those He saved? They believed in Him for saving or they would never have stepped out between those walls of water. But then they became evil and unbelieving about food and water (provision). They believed Him for a very big thing - their saving, but then they refused over and over to believe Him or obey about smaller things like food and water. So we see that He saved them but their subsequent unbelief displeased Him. This is not remaining in the trust that saved you. So it was because of unbelief that they failed to enter His rest.
And I believe that branches that don't remain in trust are in danger of being cut off. But I also see that He is very patient. He gave them 40 years before He swore they would not enter His rest for unbelief. So He is kind and merciful and patient.
So the verses in tension to the verse you give have to be taken into account too. Because if He severed a branch that did not continue in trust so that I could be grafted in, won't He sever me if I don't continue in trust? He will not treat me different will He?