There is a problem with the discussion of Hannibal's losses over the Alps. That being the presumption that Hannibal could enter Roman territory by other conventional means.

While before the Punic wars Carthage had been perhaps the preeminent naval power certainly in the western Mediterranean , Punic naval power had been destroyed by the Romans in the First Punic War. Not only had the Romans imposed severe restrictions on Carthaginian warship construction and fleet size. The results of the war had led to the Roman's having a massive fleet (as they had to build them to compete with the Carthaginians). The Romans now dominated the Western Mediterranean. Therefore any invasion by sea was out of the question.

A Land invasion from say southern Italy/Sicily was also out of the question. The these territories were under firm Roman control, and fighting in these areas would not be advantageous to Hannibal. The terrain of southern Italy and Sicily is extremely rugged, and not conducive to open field pitched battles. Throughout the entire First Punic war on Sicily there had only been two pitched battles, one of which can be dubiously called so. This is exactly the opposite of what Hannibal wants, the Romans have a Huge manpower advantage over the Carthaginians, and they have the will to use it. They will always win a war of attrition. Hannibal's elite army would be of little use sitting around blockading cities. He needed to win pitched battles and inflict heavy casualties on the Romans in order to be successful in his war.

His other option was going through Southern France by the coast, which he did follow for a chunk of his route. However the city of Massalia (modern-day Marseilles) blocked his path and was a staunch ally of Rome. Furthermore by the time Hannibal arrived in the area, the army of Publius Cornelius Scipio (headed for Spain) had landed there. Again Hannibal could not afford to waste his time sieging a city, furthermore while he could doubtless have defeated Scipio's army given his ability and numerical superiority, it would've achieved little so far from Rome's base, and either Siege or fighting Roman armies would simply give the Roman time to prepare and block his path.

Therefore the Alps offered him the best and really only option. As u/AdmiralMortarion says, the Alps while a monumental task, was not as difficult as some mountains, it was actually a quicker route than along the coast, and it allowed Hannibal to achieve his goals of taking the war to Italy.

In terms of his losses Hannibal had crossed the Rhone River with 38,000 infantry, 37 elephants, and 8,000 horse. He entered Italy with 20,000 infantry, 37 elephants and 6,000 horse. These were still substantial losses, but they were far more insignificant than they appear.Firstly this includes not only deaths, but desertions, sickness etc. While the core of Hannibal's army was elites who had served with him, his brother-in-law, and his father. This had been augmented with a huge number of fresh Spanish levies (Hannibal had raised 90,000 infantry and 10,000 horse for his attack on Saguntum and conquest of North Eastern Spain). He had already dismissed around 20,000 of these levies as unfit for his italian campaign, as well as leaving contingents in conquered areas. Therefore most of the men he lost crossing the alps were likely these Spanish levies. His best troops, including crucially his Cavalry (which were to be his decisive arm) and Elephants suffered relatively few casualties. Furthermore he entered Italy in Cisapline Gaul, a territory filled with people with a hatred of Rome ready for open revolt. So his forces were quickly brought up to strength by these Gallic Troops (his army reaches about 40,000 foot and 10,000 horse at Trasimene and Cannae).

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Sources:

Adrian Goldsworthy, Fall of Carthage 2003

John F. Lazenby, Hannibal's War 1998

Polybius, The Histories

Livy, History of Rome