I think the answer is more complicated than "I include them" or "I don't." Short answer: I don't include taxes in my day to day budget. I do include tax considerations in terms of long term expenses and allocation/investment decisions/timing.
Let's take an example: Assume your non-tax operating budget is 100K/year. Will you need to also have to account for taxes on how that 100K is sourced? Of course so. Does the tax depend on how it is sourced? Of course so. The money can come from wages (e.g. j*b), dividend income, interest income, capital gains, collectibles, tIRA/traditional 401k/403b/457 distributions, Roths, traditional pensions, social security and so on. Each of these have differing tax consequences. I would maintain that when there are a variety of choices the amount of tax you pay may be adjustable. Thus, I want to make my budget (what it takes to live each year) as independent as possible for taxes on that income. As a result, while I ALSO track expenses w/taxes, it is not my primary tracking mechanism.
For example, using the 100k expenses above. Let's say my tIRA account is doing well and will be a potential RMD landmine at some point. So, I decide to up my Roth conversions - to make it simple let's say I decide to up my Roth conversion from 0k to 150k. So now I have taxes that have to be paid on that additional 150K of income. Should that impact my thoughts/view on ongoing expenses? No.
Another approach to this is to adjust your net worth based on expected future taxes. Assume a rate on LTCG's, ordinary income, etc. Discount all of your holdings (hey, that's what spreadsheets are for) based on your expected tax impact for that item. Now, while not perfect, you can look at your overall net worth / annual operating expenses to get that ratio. Me? I don't bother (although I did do this in the distant past when I was trying to determine if I had "enough" to retire early.