Yes, it's possible using this method. This is similar to how the first fuel cell was made in 1839. You would need a lot of platinum to do it this way. Modern fuel cells use platinum nanoparticles on carbon micro-particles for very high surface area fuel cells. If you do it the modern way, you can get very high power densities. For example, the Toyota Mirai fuel cell car uses a fuel cell stack that has a power density of 3kW per liter. So, if you wanted 100kW peak power, you would need a fuel cell of this type that is 33L in volume.
This is a nice little demonstration and everything he says in the video is true. It is important to note that most fuel cells (including the one in the Toyota Mirai) only combine hydrogen and oxygen to make water and electricity. Splitting water to make hydrogen and oxygen in large amounts and for a long time requires a different setup than a "regular" fuel cell.
I worked on fuel cells for over 10 years. They are a lot of fun and can do some amazing things, and are very energy efficient, especially compared to internal combustion engines.