Why not a tax?
Taxes change value independently of the number of people using them. Therefore, if you want the revenue to be proportional to the number of people using them, you need to adjust them constantly.
Look at the gasoline tax in the United States. In many places it is intended to fund road construction, just as you would like a tax to fund broadcasting. The problem is that with more fuel efficient cars and more expensive gasoline, the gas tax revenues fell. Now, in order to get back to the previous level of funding, they either need to raise the gasoline tax (politically undesirable) or subsidize it from other revenues. Either of those is a political question and is at the mercy of the legislature.
If it were an income tax set aside, then there would be even more pressure to not spend the money or to audit its spending. Note how the federal government wants to add additional restrictions to Medicaid and food stamp funding in the US. Conservatives want work requirements while liberals want restrictions on what foods are covered.
Consider the case of a recession. Note how in the last recession, the government reduced the Social Security tax. This time they did so without also reducing expenditures to match. And they took the funds from the general budget rather than the Social Security trust fund. Part of the reason why they did this was that for many poorer people Social Security is the primary tax they pay. Briefly that was the majority of workers. It's back under 50% now but still above 40%. So no income tax reduction would reach most Social Security recipients at all (absent a negative rate).
Or consider the US shutdowns. In those, the tax would stop being collected and disbursed unless those were considered essential activities. And the designation of essential activities is under political control. Part of the reason why Germany does not have shutdowns is that their approach is separate.
Anyway, the point is that unless you are talking about a tax that is collected separately from the national tax, taxes are inherently under the jurisdiction of politicians. And thus political. Both the amount of the tax and the destination of the spending are controlled politically.
You also might consider the costs of a separate tax. If it takes three hours to fill out the tax form, which is quite possible with copying, etc. (you should make copies of your tax forms in case the government claims you filled them out differently than you did). It's easy enough to generate a cost higher than the fee.
Beyond all this, the actual reason in Germany seems to be that this started out as a fee only paid by people with receivers. As richer people were more likely to buy receivers than poorer people, that implicitly cost richer people more, even as a fee. It was only in 2013 that they changed it to cover all households.
Another issue in Germany is that their welfare system may make it easier to adjust for the fee by increasing payments to welfare. As these are generally focused on poorer households, they may better compensate for the modest cost of the fee than replacing the fee with an income tax would. This goes along with all the other costs that are more household based than income based. For example, poor people don't necessarily have lower calorie requirements, need for water, or less need for access to internet. In Germany, the welfare system is accustomed to managing such things for the poorest.
There's also the question of whether it would be appropriate for GEZ (or its replacement, the Beitragsservice) to have access to the information of an income tax. It works for broadcasters and is not under direct political control. Should people be mandated to hand over their person income data to a private organization? And should the state confirm that data (as otherwise, people could simply lie about their income)? So from that perspective, the separate structure of the organization precludes it from charging the tax (not the tax would make it political but that only political organizations can tax).
That leaves only the possibility of a state administered tax. But then the state could obviously change its auditing at any time. In theory perhaps the state could add auditing to the existing fee system, but moving collection to the state would make that much more likely. And once the state audits expenditures, it can of course influence them. Just as gas tax revenues in the US could be used for a bridge to nowhere, broadcast tax revenues could be used for politically friendly ends.