The DMG even acknowledges the game can be low or high, but neither are the standard (which is in between) by the table for Starting Equipment on page 38.The default is determined by the game itself. Even a PHB-only game is already high magic. To make a D&D game less than high magic requires limiting what’s available from the books. You’d have to remove most or all casting classes to get close to a low magic setting.
So, I don't consider the default high magic as you suggest. I know the default certainly isn't low magic! It is determined by the DM/table and how the world is presented to the players.
I would add that it is also determined by the tiers IMO for most groups. Most games never make it to tier 3, let along tier 4, so casters don't really have to be limited to make your world "low magic".
5E can certainly be high magic IF you are engaged in the higher levels at some point, give out tons of magic items, narrate your world is high magic because it is super common (streets lit by continual flame spells in any significant established community), etc.
I very much consider my game low magic, mostly because magic is rare. Casters are rare, you don't have a priest in every town to raise dead (as EGG suggested at one point IIRC...), and finding magic items, especially permanent ones are a true treasure!