Basic answer is that the 5351 has higher phase noise (jitter).
ITs adequate for HF use but at higher frequencies where the ambient
and galactic noise are lower it handicaps the receiver performance.
This is old news.

The Si570 is roughly 10x lower jitter ans as a result less phase noise.

Compared to a good crystal or VFO they are an order of magnitude nosier.
The price paid for stability and digital tuning.

--
Allison
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Please use the forum, offline and private will go to bit bucket.

I appreciate the help offered by every one who has explained the theory and suggested methods of calibration and ways to verify it.

It looks like I have succeeded in improving things, with minimal test equipment. (However, I will investigate the test devices which you have suggested.)

The diagram from Evan is helpful, because it roughly corresponds with what I see, now that things are nearly where they should be. See my diagram below. This was drawn from two separate spectrum plots, as done with an Antuino.
I expected to see a much narrower BFO clock, which would be positioned near the upper end of the crystal filter passband. (Below the diagram, my comments continue.)

Further comments:
The HFsignals online BFO calibration tool is helpful for getting things aligned well enough to receive something. However, as I see it, it is rather imprecise and subjective. The positioning of the noise pedestals is subject to the user's judgment. The background noise may differ substantially in various regions or environments. I could not get a well defined pedestal at the low end of the spectrum. When I set things as closely as I could, to match the online demonstration, my radio was not differentiating sidebands, and was transmitting RF in the absence of audio modulation. Now that things seem to be tuned correctly, the displayed noise is much stronger in the mid to high end of the 300 - 3000 Hertz range than it appears in Ashhar's demo.

With the help which you all offered, and with hints found elsewhere on the forum, here's what I did:
1. Perform the normal BFO calibration (with antenna connected, and with receiver tuned to noise, not signal).
2. Adjust BFO calibration while listening to an SSB signal, for best audio clarity. Start with a USB signal. Repeat with an LSB signal. (If using KD8CEC firmware, ensure IF Shift function is Off.)
3. Calibrate the receiver frequency, to a known strong signal... Preferably, one of the WWV frequencies (2.5, 5, 10, 15, or 20 Mhz). However, the WWV signals have a long way to come to reach me in Alaska. Therefore, multiple skip paths often confuse attempts to find zero-beat. Occasionally, I have found a strong HF broadcast signal, without much fade. Then the trick is to know the station frequency. (I have been advised to buy a 10 Mhz frequency standard, to avoid propagation issues. But that has yet to happen.)
4. Recheck BFO calibration (step 2.)
5. Adjust CW BFO calibration (not on all systems, but included in KD8CEC firmware). I have not yet done this. But I think it involves listening for best reception of CW signals in both CWU and CWL (as was done for SSB, in step 2).?
6. Verify correct sideband. A few other posts suggested tuning to 7074 Khz in the evening in USB mode. If propagation is good, you should hear FT8 signals. You should hear nothing in LSB mode. If signals are present in LSB mode (not USB), the sidebands are reversed. Using the same procedure, verify CWL and CWU are not reversed (signal should be heard only in CWU).
7. With the microphone disconnected, transmit SSB into a dummy load and monitor power output. (Key the transmitter from the mic jack, not from the keyer jack). There should be no output power. Check both LSB and USB.

Hopefully, that should do it. (I have put the original BAT45 diode back in place.)
Thank you all.