Those "savings", if they are correct (serious doubts have been raised about that) looks to me like a set of short-term gains for long-term pains.
It's like if I stop commuting by car to work every day and rent a helicopter. I'm going to save a ton of money by selling the car and not buying gasoline. Of course I spent a ton of money with the helicopter method but we're not going to list _new_ expenses on the website are we?
Spending massively more than you make is just fine, until the repo man shows up.
Tax enforcement is a trivial and almost immediate example. Spending $X on compliance seems to recover about $5X in evaded taxes. On top of that, there are knock-on effects: if it becomes easier to cheat on your taxes, more people may cheat.
Vaccines and other forms of preventative healthcare fall into this bucket too. Even completely ignoring the moral aspect of letting people (mostly kids!) fall unnecessarily ill, it often makes economic sense to pay a little bit to avoid having to potentially pay much more down the road. One ER visit can cover a lot of flu or COVID vaccines; a few nights in the ICU even more.
Research grants are maybe less obvious, but they have a huge multiplier too: the human genome project had something like a $120x return(!). This is not just big breakthroughs, but also all the work along the way. A lot of grant money goes to training people or supports small businesses making their equipment. I saw an interesting article claiming that the Air Force essentially bootstrapped the use of more exotic materials: military contracts covered the initial investment in (e.g.,) machinery for working with titanium, and once those fixed costs are covered, it was feasible to dip a toe into the consumer space and see if there's demand. Thus, titanium golf drivers.
There are many documented instances of DOGE cutting things that they later realized were needed -- leading to unnecessary switching costs and other consequential costs.
Second: deferring costs in the short term are often a bad choice that can cause higher costs over the long term.
Third: some cuts can exchange monetary costs for non-monetary costs. These will make a number look good but can cause impacts that are bad.
Junk food can be cheaper than healthy food, it's not good to save on money now at the cost of spending thousands on medical care and insulin in the future.
Or to paraphrase Terry Pratchett: "boots at $10 dollars that last a month are more expensive than $50 dollar boots that last a year" https://en.wikipedia.org/wiki/Boots_theory
It's the difference between being cheap and being frugal...
There's also the allegation that Elon's Doge cuts were to prevent various agencies from pursuing legal action against him.
But often, "you get what you pay for". I addressed that also in my short comment.
For these savings to be good, they need to be good for everyone. And that's not clear.
[1] https://www.unitedformedicalresearch.org/annual-economic-rep...
If we as a society choose to stop investing in ourselves, we'll have bad outcomes in the end.