Excellent question! Very insightful.
As a native English speaker myself, who has been a native speaker in four 'English as the official language' countries, (South Africa, New Zealand, Australia, England) I can honestly say that even in an English as a first language country vernacular is more a thing of culture than language.
I recently learned a lesson at work here in England; I said to my colleague about a particularly difficult manager and policy "I really had to put on my 'big boy pants' today"
For those not in the know, 'Big boy pants' is the way of saying that you have to behave like an adult. And in NZ, AU and ZA this would be a casual statement and would pass without anyone caring. In England however, "pants" is the word for Underwear, so this statement did not pass by quite so casually.
Now the OP's question really has a great truth to it, should a speaker in the workplace use words that could be taken out of context? (even if the chance of it being so is small), Should a speaker give extra maybe even excessive consideration to the words and phrases that they use when speaking?
Could a word like "religiously" be detrimental to the speakers ability to communicate in a clear and understandable manner, without offence or misunderstanding?
I think this applies to all level of business not just someone who is speaking to a thousand people at a time.
A good communicator is someone who is able to clearly communicate an idea, without bias, in a clear and concise phrasing, that as many persons listening as possible will understand and absorb without the listener needing to interpret the message into digestible language.
There's some serious physiology and psychology behind how humans process sounds and language in particular, although I can't find any credible references at this point in time. When listening we interpret the words from their complex forms, down to simple forms that we then understand as the sentence spoke, such as "can't" is resolved back to "can not" and "religiously" is resolved back to "every day all the time" (depending on your interpretation)
(Recommend this excellent bit of reading https://www.mtholyoke.edu/acad/intrel/speech/differences.htm )
So no, you should not use a phrase to represent an idea, because a phrase that represents an idea, requires more processing by our mind, and detracts from the meaning of the words being spoken.
The fact that the word 'religiously' has a religious/cultural/social tone of it's own, is is also a valid point to bring up, mainly because as a speaker you do not know if someone will interpret the meaning of the phrase to be the same as the meaning you intended.
So no, you can not expect a certain level of understanding of the vernacular in a corporate environment, because the words you're asking about are not from that corporate environment, but have a cultural basis that requires the listener to have the same cultural point of view as the speaker.
And lastly the question of causing offence.
"Better safe than sorry" to use a phrase.