When can “that” be replaced with an -ing form

Nowadays is a complex temporal adverb that means much the same thing as Positive anymore.
That is, Nowadays, S asserts that S is true at the present time,
and also presupposes that there was a time in the past when S was false.

• Nowadays, everybody carries a phone.
• Everybody carries a phone nowadays.
• *%Any more, everybody carries a phone. *
• *%Everybody carries a phone any more. *
  (The '%' in front of the sentence means it's dialectal -- see the link for details)

This is the opposite of normal any more -- which is a Negative Polarity Item
and therefore can appear only in a negative clause.
Normally, Not S any more asserts that S is false in the present,
and presupposes that S was true in the past.

• Nobody wears a fedora any more.
• Nobody wears a fedora nowadays.

But the reason why the second sentence is preferable to the first has nothing to do with nowadays.
Start is an active inchoative predicate, and therefore its use in the present tense
is interpreted in a Generic sense. With a punctual predicate like start,
that would mean a repetitive sense, as in

• The motor starts to sputter every time I press the accelerator.

That is, the "present tense" starts to get is not used to refer to the present time;
the progressive is starting to is used instead.

To narrow in the definition on the specific example you gave:

Catering in this school has been franchised to the company.

It implies that the company operates the business, but that the school still controls the brand.

The company may be quite a large one, that runs cafeteria and catering services in many institutions. But the implication of franchise is that the operator is working under the school's name (or the name of a brand the school owns). If you go to a cafeteria, it will be identified as the "Brown Residence Cafeteria" or something similar, not as "Company X Food Services".

Many people would consider this type of franchising (of an institution's activity to a large operating company) a type of outsourcing, but not all outsourcing arrangements would be considered franchises.

Of course, as the other answers emphasize, there may be other important distinctions when using the word in a legal sense.