Saving money is generally seen as a good thing.
Moderation (in spending) helps preserve wealth, whereas excessive consumerism leads to wastefulness and the loss of wealth.
In reality, it is more productive to focus on new ways to earn money than to try to save.
Spend less, keep more and you have a better chance of staying financially stable
There are some nuances. For instance, in what form will you be saving? If you save in traditional currency, then unfortunately, you will lose more than you save (due to inflation).
but I start to think about what happens when a lot of people start doing the same thing at the same time.
People start cutting back en masse during tough times, when inflation "eats up" a large portion of the budget due to rising prices for goods and services.
If people become very careful with their spending then businesses sell less.
Of course, falling, but inflation (traditional economic model) is to blame for this, not people's frugality.
Some businesses may respond by cutting cost or reduce staffs.
Because, just like ordinary people, they are trying to "survive" and optimize their business (losses, expenses).
That can then affect people’s incomes and make them even more cautious about spending.
Yes, in this world, we are all economically interconnected. A "difficult financial situation" in one sector inevitably ripples out to others, affecting ordinary people along the way.
What makes it tricky is that nobody is necessarily doing anything wrong. For an individual, saving more during uncertain times can be a smarter decision but if millions of people make that same decision at once, the effect on the economy could be different.
So, what are you proposing? That millions of people start increasing their spending, using borrowed money, say, to save the economy?

I repeat: everything is interconnected. I would also add that this is simply how the current global economic model is structured, a model that will likely undergo some changes in the future. For instance, "expiring" digital cryptocurrencies (CBDC) could be used to artificially and forcibly stimulate the economy by compelling consumers to raise or lower their consumption at specific times.
Which brings me to this question; can something that is financially responsible for an individual become harmful when almost everyone does it at the same time?
Your question is overly philosophical. Any action inevitably entails both harm and benefit, whether to the individual or to others. That is simply how the world works. So, what would you choose: the survival of your family, or benefiting the national economy?
Consider another example: you buy an asset (bitcoin), hold it for ten years while it appreciates, and then sell it for ten times the original price. By doing so, are you inflicting financial harm on the new buyer? That is the nature of the market - and of life itself: there is always a winner and a loser, and the weak (be they individual market participants or businesses) lose out and "die off".