Learn English – Beginning a paragraph with Also or Furthermore

There seems to be some evidence that as of late (in the sense of recently) was perhaps a bit more common a century or two ago, but I think including as has never been the norm. For all that, I can't really imagine a context where OP's example...

"and as of late he has agreed to return home"

...would be considered "natural", even if we removed as. The reason it doesn't work is because of late in such contexts means during the time period stretching from the present moment to the relatively recent past. That's to say, things happening of late are normally things which either take some time to complete, or which can happen repeatedly (unlike, say, a single act of agreement).

As @Edwin succinctly notes, lately/of late apply to durative (rather than "punctive") recent actions.

In OP's case, the natural phrasing would be...

"and he [has] recently agreed to return home"

There are several meanings for the word “sentence”. The online Merriam-Webster describes “sentence” as “a group of words that expresses a statement, question, command, or wish”, without restricting its grammatical form. But it also describes “sentence fragment” as “a group of words that is written out as a sentence but that lacks a subject or verb” (as the short version of the definition), thereby implying that a sentence proper has both a subject and a verb (predicate).

In a more grammatical approach, it is the presence of a predicate that makes a sentence. Many languages allow the omission of a subject, since the agent is expressed by the inflected form of the predicate. Even though this might be described so that the subject is present, but implicit, there are also many languages where sentences with no subject are normal, e.g. sentences corresponding to “It is raining” do not require a formal subject.

It is still a matter of definition what you call a sentence. And it is anyway a fact that people frequently use expressions that are not sentences in the grammatical meaning mentioned above (i.e. lack a predicate), yet serve the function of such a sentence. They are very common in short notices, news headings, etc. Usually the predicate can easily be inferred, as regards to the meaning, though we might have different ideas of the exact wording. But the point is that such wording does not matter.

“System shutdown in 60 seconds” can be read as “System shutdown will take place in 60 seconds” or in a few other ways, but the meaning is the same. Omission of the predicate probably makes communication more efficient here: the message is shorter and can thus be read and grasped faster.

However, such expressions are not accepted in normal flow of written nonfiction prose, or in “formal prose” as some people might say. The reason is that such forms of language have been developed to have a certain structure, including “complete sentences”, i.e. sentences in the grammatical sense. This makes them more readable, since this is the form we are used to expect. When you use sentence fragments, you break the normal flow. This may be an important technique in narrative text and even in nonfiction prose at times. But when used just because you don’t care to formulate your messages as normal prose, it makes the impression of… lack of care.

This is why Microsoft Office Word shows an error message in Spelling & Grammar checks, if you try to use such expressions. This can be changed by editing the settings. The way to do this depends on Word version, but the setting in Proofing options is “Fragments and Run-ons”. Even if this setting is checked (as it normally should), Word accepts fragments in many contexts. For example, if you write “Hello world” as a heading, or as paragraph of its own, it will pass. We can say that Word expects it to be a title or something like that and checks just the spelling, not the grammar. But if you end with a period, Word interprets that it is meant to be a sentence and says: “Fragment (consider revising)”.