First of all, thanks very much for the incredibly helpful responses, those got me on the right track, though I'm not sure what I did is OK...
Doublecheck your opamp is properly soldered to the breakoutboard.. Sometimes happens it gets rotated (pin1 is pin5 then).
And that is the opamp's output which creates the -1V then at the divider..
Yes that happens and I can appreciate why. It's easy not to see the tiny marks indicating pin 1. Even sillier I had killed my first opamp in this prototype by wiring it as a single AD8033 when in fact is/was a dual AD8034.
You are turning on an unusual feature in this amp as the frequency goes up. If you look at the CMVR, it has a FET input range but with a bipolar across it for overloads. The bias current is a pico Amp for the FET range but then goes to nano amps when you exceed about 2v on the input. At high frequencies, the caps on your dividers are shorting and high CMVR gets to input. All your coupling caps get charged up due to the higher current and the opamp dutifully follows. In the last iteration, you AC coupled + input, this won't work, you need to have a DC path. I guess the thing to do would be to bias the amp with a divider and then pass your signal in.
That was very helpful, thank you. I wired the opamp as a non-inverting amplifier and two 2k resistors for the feedback voltage divider. I had read (here
https://www.renesas.com/en/document/apn/r13an0003-how-bias-op-amps-correctly) the input needs an DC path for the input. Like this:

This worked. However, that input resistor and capacitor form a very well working high-pass filter, so with sine wave input it worked only at 5MHz. With a square wave input it worked down to 50kHz because of a short spike that the comparator (which follows the opamp) was picking up.
I tried removing the entire filter while leaving the amplifier feedback intakt, and voila, that does the trick:

This appears to solve the voltage offset problem this thread started with. The opamp the output is "malformed" in the negative, not sure why that is. In my case that's not a problem as the comparator just needs to see the negative half-wave, it doesn't really matter how well-formed it is.

So the whole circuit now seems to work even with a small 0-3V input signal (my SI5351 frequency generator doesn't do more than 3V), well enough to work to at least 25MHz.

TP12 is not a good probe point- this is an ultra sensitive node.
Indeed, for the tests today I attached the probe briefly just for verifying the signal was there, but otherwise detached it. For comparison, the output changes significantly whether or not the probe is attached.


Unfortunately I have no good alternative point for probing the opamp input.
Frequency Counter inputs are generally pretty sensitive, maybe a 1/2v max. Often they'll have a 20 db atten to knock down signals. Reduce your signals a bit.
I was under the assumption that's what I'm doing, but I have since learned, I'm not. I guess this is something I have to figure out because I have a feeling the circuit will unduly strain the signal source, which is a test point in my HP3310A frequency generator and not the high-power output.
I suppose things should improve if R8 and R9 are reduced. And I'm not sure if R9/C18 are useful at all?
And why are the ratios of R7/R8 and C17/C16 different? This means different attenuation at different frequencies.
That was a mistake on my part, I changed the capacitors to 10p and 68p to match the resistor ratio exactly. I didn't try (yet) to reduce the resistor values because I'm trying to put as little load as possible on the signal source - though I'm no longer sure that is the way to go.
Thanks again for all the helpful advice!