Learn English – agreed helping him v/s agreed to help him

Your first example is idiomatic; your second is not.

"Late getting back" is an elision of "late in getting back," where the gerund is used as complement to the preposition. Here, "in" carries a meaning something like "related to."

I was late in realizing that he was secretly a werewolf.

"To" does not have this meaning. The preposition "to" is used with "late" only where the complement is a noun:

I was late to the party.

I was late to my own wedding.

or with the adverb "too" to indicate that the lateness was excessive:

I was too late to save her.

He arrived too late to meet the famous molecatcher.

"So" does not work in these sentences the way that "too" does, so your second example would not be idiomatic in most English dialects.

Here is what you are looking for, both 'have to' and 'having to' give the same type of sentences but the event/state in which both sentences have been said is important.

"Have to": to need to do something.(obligation)

"Having to" is the progressive form. (obligation in progressive state)

2. People are having to leave their rural areas.
3. She is having to look after herself now.

Both sentences are in continuous tense with the obligation expressed by having to