There are several issues here, and addressing them should make the picture more clear.

First, Portuguese going down Africa's coast was at first motivated primarily by their wish to reach the source of goods arriving through the Trans-Saharan trade, coupled with their military-political-religious objectives to conquer Morocco. It is only after the success and profitability of this African trade that the main objective became reaching India. I talked a bit about Portuguese start of discoveries and their motivation here in my old answer here

Second thing is the general advantage of maritime transport over overland routes. By all metrics shipping goods by ships over sea is more effective than over land. It's true today, and it was especially true back then. It is faster, cheaper, less labor intensive, and allows larger quantities to be sent. While a single ship of the time could transport anywhere from 50 tons (small caravels) to over 1000 tons (largest carracks used by Portuguese) of goods, carrying just a fraction of that amount of cargo by land would necessitate a huge number of pack animals, people to tend to them, soldiers to protect from bandits, all of which translates to huge expenses.

Third is, that by the time of discoveries the overland "Silk road" has fallen into decay and disuse from its heyday in the time of Mongol Empire when people like Polo used it to reach China. Simply put, safety to use the route drastically decreased, while new political entities further made the route unprofitable as each charged their own customs, taxes, and duties on the goods, increasing the price. If they would even let the goods pass. These challenges weren't there just for Europeans. Already by the fifteenth century in Asia, the overland route through Central Asia was replaced by seagoing route starting in China, going across SE Asia to India and then ending at Red Sea and Persian Gulf. I talk more about the routes here

All in all, Portuguese were quite distant from the terminus of the Silk Road at the Black Sea, and as a minor point, I don't see how they could even reach it without going through the Mediterranean. From there they would have to use a now dangerous and expensive overland route to travel if they would even be allowed to pass. And then, they would be met with an insurmountable challenge to move huge quantities of goods by these overland routes through many foreign lands. Compared to that, going by ships would entail putting huge quantities of goods on the ship at the source, travelling on open sea by the most efficient, fast and cost-effective means of transport at the time, and unloading at the home port which also meant they only had to pay taxes duties in the source port and at their own, drastically reducing the cost.