It’s kind of spinny, but I don’t think it’s necessarily false. (I’m assuming the numbers in the fact sheet are honest as far as they go and don’t actually misrepresent the subset of data that they’re talking about.) Nor does it contradict the claim that most of the tax-cut benefit goes to upper-income taxpayers. It’s a question of amount of change in tax rate vs. amount of change in actual taxes paid.
Here’s a vastly oversimplified example of how this seeming paradox works. Let’s say you and I constitute the nation’s taxpayers; I have a taxable income of $10,000 and you have one of $100,000. I’m taxed at the rate of 10% of my total income (let’s leave marginal rates and brackets out of it for now), and you’re taxed at 34%. That means I contribute $1000 to the total tax revenue while you contribute $34,000: out of the total tax revenue of $35,000, I’m paying a little under 3% (2.86%, if you care) while you’re paying a little over 97%.
Now let’s assume that we each get our tax rate cut by four percentage points. I now pay only 6% of $10,000 ($600) in taxes while you’re down to 30% of $100,000 ($30,000). My share of our combined tax revenue of $30,600 is now only 1.96%, while you’re paying over 98%.
Now, you got a tax savings of $4000 on this tax break while I saved only $400, so it’s undeniable that most of the benefit of the tax cut is going to you. But since my tax rate and my income are so much less to begin with, the identical rate cuts mean that the total revenue ends up being even more heavily weighted towards your contribution.
If we wanted to cut your rate by four percentage points and still have me end up paying the original 2.86% of the total tax revenue, my rate cut would have to go from the original 10% to 8.83%, which looks pretty measly and nets me a tax savings of only $117.
In short: the more income inequality we have and the more progressive our tax system is, then tax cuts for all tend to skew the percentage of total revenue burden ever further towards the higher-income taxpayers. Consider the outcome if we both got a seven-point rate cut: I’d be contributing a mere 1.09% of total revenue while you’d be forking out an astronomical 98.9%! What a lazy freeloading scum I am, huh?
Now, if we stop and think about it, we probably all agree that in the present economic climate, a tax burden of $27,000 on someone with an income of $100,000 doesn’t actually entail nearly ninety-nine times the financial hardship of a tax burden of $300 on someone with an income of $10,000. That’s why progressive taxation exists: because rates and shares and percentages aren’t very meaningful if considered in isolation from the absolute numbers of basic financial survival.
But for the hard-line anti-tax crowd, cherry-picking the numbers that way is pure PR gold: basically, it means that the more you cut taxes, the bigger the burden on the richest taxpayers appears to be. Makes you feel kinda sorry for them, doesn’t it? I know I feel a little guilty looking at my lame-ass 1.09% donation and thinking about your staggering along under the remaining 98.9%. Hey, I know: why don’t we raise tax rates for all a couple of percentage points instead of cutting them, so that I pay 12% of income and you pay 36%? That way I’ll be bearing a much more responsible 3.22% of the total tax burden! Don’t you feel happier? 
(Then there is the question of the “spinniness” of comparing share of total tax burden based only on federal income taxes rather than including payroll taxes, which are much more heavily weighted towards the lower-income taxpayers, but let’s not get into that now.)