Do Pokemon have default move sets

It's disadvantageous to stick to a single Pokémon, but not all that impractical. There was once a run of Pokémon Yellow, I believe, done with just a Pidgey.

You don't need multiple high level Pokémon, it simply helps. Type coverage is the main reason - for example if your highest level Pokémon is just a Fire type and can only do Normal and Fire attacks, then you stand a hefty disadvantage against Rock types who resist all your attacks and can deal double damage. As well, Fire and Dragon types will block you completely, while Water types mostly block you and also do double damage.

Having your other Pokémon be of equal or near level as your highest helps in survivability of that other Pokémon as well. For example, a Ground type with high defense like Sandslash can help protect against the Rock types while also taking them down. This also gives you the opening to fully heal your primary Pokémon in between the assault of your opponents - it's a lot easier to keep your primary Pokémon in top shape when your supporting ones don't fall down in a single hit. At this stage, though, it probably helps to make those "assist Pokémon" simply capable of taking down foes on their own right.

It's ultimately up to the preference of the player, though. It's fairly common for people to stick solely with their starter of choice, and by sheer force of their levels take out all opposition. As long as you have good type coverage with it (for example, Charizard can use the TM to learn Earthquake to help mitigate some of the problem foes), enough of a level advantage will keep you on top.

Bulbapedia indicates that Blizzard and Ice Beam can not be learned by a gen 1 Blastoise by levelling up, at all. It can only be taught those moves by using a TM.

There are some pokemon who learn a move by levelling up in some of their basic evolutions that they cannot learn in a higher evolution. For example Raichu cannot learn Slam, but it can have the move if you levelled it up to level 20 as a pikachu before evolving. This is the closest equivalent to that which you describe.