What does the word all include?

We learned at the beginning of Tractate Minachot:
 
All offerings that were not cut into pieces – are considered worthy, but did not go up to the owner for a duty, except for a sin offering and a zealous offering.
The Gemara on page 4a cites the opinion of Rav that even an Omer offering that is not for its own sake is invalid because it was intended to permit and did not permit it. The Gemara asks why the Mishnah did not specify: “Except for a sinner’s offering, an offering of zeal, and an Omer offering”?
The Gemara answers:
Because I am a sheaf – an individual comes, a group comes, not a sheaf; because I give – it comes for its own sake, it comes because of a sacrifice, not a sheaf; because I am a sheaf – you see, there is no fixed time for them, there is no fixed time for them, not a sheaf. “The offering of the Omer is not like other offerings because it is a public offering, etc.
What is the meaning of the Gemara’s excuse? Does it mean that the phrase “all offerings” at the beginning of the Mishnah includes only individual offerings for which there is no fixed time, and that public offerings are not included in “all offerings” at all? If so, why did the Mishnah not limit the generality?

It could perhaps also be said that the Mishnah did not lack external expression, but rather individual offerings, but not public offerings, and again we need to understand why the Mishnah did not emphasize this.

Thanks in advance.