The sentence you didn't hear properly is:
Sunday lunch is not simply about refueling but a relaxed communal experience centring on a well cooked meal.
A native speaker would not use centric on instead of centring on in the sentence. Secondly, the words are not synonyms unless they have or are used as the same part of speech.
Centric is an adjective. Centre (US: center) is a verb. It is also a noun. And nouns can be used as adjectives. Rarely, adjectives can be used as nouns (the poor). Therefore, unless the two words are being used as the same part of speech, they are not synonyms.
As a verb center/centre is quite often followed by on. See center on and the example sentences.
Since centre on can be part of a verb + particle combination, the use as a gerund centring on is not surprising.
We use gerunds to modify nouns all the time. You can also use a past particle as an adjective, as in:
Sunday lunch is not simply about refueling but a relaxed communal experience centred on a well cooked meal.
One can use an adjective instead of the gerund (centring) or past particple (centred):
Sunday lunch is not simply about refueling but a relaxed communal experience central to the day.
Central is a much more common adjective than centric. The use of centric to and centric on is hard to find. Central to is the overwhelming usage.
You can find examples of centric on using a google search in goole books, but almost all of them are false positives, such as
Turner mixes basketball and soul music with a James Brown-centric on-air TV campaign to hype the playoffs.
As for more on gerunds, the use of centric as an adjective, and how that is synonymous to the use of -centric, feel free to read the following.
Sitting around the Irving Street kitchen table, they discussed rewriting the winning blueprint they'd devised thirteen years earlier. "Instead of a single recipe," he said, "why don't we do some menus building up to a party?"
Notice the noun menus followed by building up to a party. If one wanted, one could easily use centring on/around a party here.
Why don't we do some menus centring on/around a party?
The next sentence of my source is
Julia loved the idea: a "meal-centric" program for occasions, like birthdays.
Here, the the adjective centric is being used with meal to form the compound adjective meal-centric. A hyphen is often used when two adjectives are used as a compound adjective to desribe a noun, here program. Of course, meal is normally a noun, but like most nouns can be used as an adjective, as in meal ticket.
And this use of centric is synonymous to the suffix -centric, meaning "having a specified centre. A meal-centric program is a program whose specific center is that of a meal.