First, ROCK ON! for building an HF receiver from scratch.
Second: short answer, a passive tunable LPF with a knob can be effective, and is a time-tested approach. (Look at lots of older receivers for a knob labeled "preselector" or "resonator.")
More Detailed Answer:
Why is there an LNA between the antenna and the first filter?
For an HF receiver, where there is such an abundance of man-made noise, the LNA at the front end is unlikely to be of much use. But more importantly, the gain element functions as a mixer. This mixer creates intermodulation products from unwanted stations beating against each other. This can put more than images into the IF passband. That bare LNA is likely to cause more grief than pleasure.
For the same reason, you may find varactor tuning in the front end problematic. There probably are HF receivers with varactor tuned front-ends, but passives are a good first-line-of-defense. That junction in the varactor just begs to become a mixer.
For an example, I looked back at a venerable (and perhaps revered) TRX design from the 70's -- the TenTec Triton IV. It had a preselector before the gate of the first RF amp. The inductor was slug-tuned (!) from the "resonator" knob on the front panel. I notice that the Signal/One CX7 also had a passive preselector before the first RF amp. (The latter design is really something -- I owned one back in the 80's and have regretted selling it ever since.)
At a minimum, you'd benefit from a high-pass filter with a cutoff around 3MHz if you are in an area with broadcast stations in the 540-1600 kHz region... And with any kind of broadband gain near the antenna, a low pass with a cutoff between 13 and 50MHz would be a good idea too. Intermod from FM broadcast stations can be particularly noxious as the signals start out very wide.
The scheme you have looks like you can move some things around and experiment with the gain distribution/budget. This is a really big deal, as not all gain is good gain. You might also try alternative mixers for the first mixer. The AD831 looks really cool, and has pretty good IP3 specs. I wonder how it would compare to a passive mixer like the TUF-1 or SBL-1 (perhaps there's a museum near you that has one
) The latter are passive, and are more "bullet resistant." (This is nothing but a suspicion, but I wonder how the 17dB difference in LO drive level affects spurious responses. The AD831 requires -10dBm of LO drive, while the SBL-1 requires 7dBm. Imagine a spurious tone (say from the BFO) that couples to the 831's LO input at -50dBm. That's only 40dB down from the real LO. But for the SBL-1, that same spur is 57dB down. Like I said, this is only a feeling, not the result of careful study or experiment. )
Again, good-on-ya for building an HF receiver.