I understand that while supplemental income is ultimately taxed the same as other kinds of income, it's withheld at a flat rate of 22%.
If this causes a significant under-withholding, is the employee penalized?
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To answer the question, yes it could result in underpayment penalties. There is no exemption from penalties for supplemental income.
An easy fix is to update your W-4 to withhold additional tax for the rest of the year, but that requires estimating your total tax bill for the year, calculating how much will be withheld at the current rate, and determining how far short you will be.
The U.S. tax system is a pay-as-you-go system. If you are getting this income, you are expected to pay taxes on it. If not enough was withheld, you are expected to be making at least quarterly estimated tax payments.
In terms of penalties, if in the previous tax year you had withheld enough, most people would qualify for one year of 'safe harbor' wherein the penalties for withholding too little are waived. However, you don't get two years; you are expected to have learned from the previous year and either adjust your withholding forms or again be making proper estimated quarterly payments. The details on what exactly is 'enough' are a little complicated, a good place to start may be here: https://www.irs.gov/taxtopics/tc306
As you said in the end, it doesn't make a difference. Any overage or under amount is generally not a big deal, unless your are very close to hitting the under-withholding penalty.
Assume a single person in 2019, and the bonus is $1,000.
If they make $1,000 less than the top of the 12% bracket, then the entire bonus would fit within the 12% bracket. So if they make less than $38,476 they would have has $220 withheld due to the supplemental rule, instead of the $120 they would have owed. Therefore having a bonus will not have too little withheld.
If they are in the 22% bracket ($39,476 to $84,200) then the supplemental amount withheld ($220) matches the amount they will owe.
If they are in the 24% bracket ($84,201 to $160,725) then they will have too little withheld. They will have $220 withheld instead of the $240 they owe for the bonus.
if they get to the 32% bracket ($160,726 to $204,100), then the difference is bigger. Again $220 is withheld instead of $320.
Now if the bonus is $10,000 the numbers get bigger.
If the employee sees that this will be a problem there are a few options.
For example a bi-weekly check of $1500 will have the last dollar withheld at 12% but that extra large check with the $1,000 bonus the last dollar will be withheld at %22. But if the bonus was $10,000 the last dollar in the $11,500 paycheck will be withheld at 35%.
So if you are going to be close, and want to avoid a problem: have extra withheld, or make sure you make the safe-harbor numbers.