Sure, computers could solve this problem, but the human cost—changing accounting systems, communicating new dates, etc.—is not negligible. A simpler (and probably cheaper) solution might be to update the IRS systems to scale better for the week before and after tax day.
Because IRS doesn't write the law, that’s Congress's job, and the tax filing deadlines (and the other deadlines that are coordinated relative to them, like the deadline for filing and supplying to taxpayers W-2s, 1099s, etc.) are set in law.
As to why Congress doesn't do it, well, the fact that their are coordinated deadlines for multiple linked processes that different parties have to comply with would produce chaos if 1/12 of taxpayers had a tax deadline each month, and all the linked forms for those taxpayers, and the applicable tax years, had similarly staggered dates. OTOH, if you just gave some taxpayers longer after the availability of forms and end of the (single, consistent) tax year to file, there would be substantial fairness issues.
But be careful what you wish for. The IRS could decide you have to pay your taxes by midnight on your birthday (because it’s easy for you to remember) and then where would you be?
That’s right. Drunk and filing your taxes.
So, you could give everyone a different tax year. The forms already allow you to have a different tax year, though few people make use of that (I have no idea what qualifies people to have a different fiscal year, it might just be for companies, I don't know).
Of course, this would be a giant mess because all the other entities that are tied to the standard tax year, namely everyone who communicates info to the IRS about you and files 1099-xyz forms about you and sends copies to you: your employer, your bank, your IRA/401k company, etc. I don't see how it'd be workable at all.
Here's what would work much better: simplify the tax code (eliminating a lot of deductions), and make filing automatic for regular W-2 wage-earners, like they do in western European nations, where they just send you a pre-filled form, you make any corrections to it (to account for income they didn't know about), and then send it back if there's changes (otherwise they just go with what they already know about you). We don't do this in the US because we're corrupt: the tax-filing companies have bought off the politicians so they won't let the IRS do this.
It's mostly for businesses (which includes individual tax payers as sole proprietorships), and the things looked to for non-automatic approvals are things like what your actual cycle of business is like. (For individual tax payers, the only automatically allowed changes are, IIRC, change back to a calendar year as tax yesr and changes to align with a spouses tax year with certain conditions.)
https://turbotax.intuit.com/tax-tips/small-business-taxes/s-...
Sounds quite complex. I'm surprised they permit it.