He may be right. The income tax tends to be more stable than a consumption tax in an economic downturn. But income taxes discourage specialization. When it costs you 2x what you make to hire someone else, then you do it yourself.
Hence the growth of Home Depot and Lowes.
If it is relatively flat, yes. But, if it is progressive, it can be volatile.
That's one of the reason that federal government revenues crashed in 2008. The rates had been adjusted to the point that the bottom 50% paid almost no income tax, and the top 1% were paying a bigger proportion of federal taxes than ever before.
But, the income of the top 1% is variable, and much less stable than the middle class. The income (especially from capital gains) dropped off, and so did tax revenue. And it made the deficit that much larger.