Here are some of the reasons that focusing on the breath is beneficial in meditation.
Using the breath: Why? Assuming you see the value in taking up a
specific object for developing concentration, the breath has many
things to recommend itself as the object of choice. 1 - it is
portable. Every where you go you have it with you. No need to worry
about forgetting your worry beads. 2 - it comes free with every body.
No need to buy any special equipment. 3 - it is complete in and of
itself. No need for any upgrades or add-ons. 4 - it is 100% natural -
they don't come more organic than this. 5 - it is effortless. The body
knows how to breathe without you needing to do anything, You just sit
back and let it do all the work - while you just watch. 6 - it is a
connection with a vital life force. 7- it is calming. There is a
simple, natural rhythm the breath follows and following that leads one
All of these reasons but especially 4, 5, 6, and 7 indicate that breathing should be natural and unforced, effortless and done in a relaxed way. So the emphasis is not on trying to make breathing to happen in a certain way, but allowing it to happen natural. If the nose is blocked from mucous then the mouth is the only option.
But on the same page are many variations of how breath can be integrated in meditation:
examples are counting, following, listening to noise. Observing and turning away.
Walking meditation can also use breathing in it.