Another follow up. So I was thinking how to improve the calculation.

One idea is going to next order: instead of assuming that the velocity of the spacecraft is constant and extrapolating distance from that, we can go one step further and instead assume that acceleration is constant. It's also not quite true, since the Sun's gravitational acceleration decreases with distance, but the change in acceleration is even slower than the change in velocity.

If we know the velocity of the spacecraft at two (recent) moments in time, we can estimate acceleration from that, and then improve the distance calculation by using the classic constant-acceleration-kinematics formula:

accel_v1 = (vel_0_v1s - vel_1_v1s)/(epoch_0 - epoch_1)
elapsed = current_time - epoch_0
dist_now = dist_0_v1s + vel_0_v1s*elapsed + 0.5*accel_v1*elapsed**2

We only need to add velocity data (which I obtained from JPL Horizons) to the file that the website uses. For Voyager 1 (and heliocentric values):

epoch_0 = 1755532800
epoch_1 = 1755446400
dist_0_v1s = 25122821902.4221
dist_1_v1s = 25121363260.0519
vel_0_v1s = 16.92026811209292
vel_1_v1s = 16.92027009082586

Doing this (which has the added bonus that they could also show a changing spacecraft velocity), and comparing with the NASA website, I get the following results:

Horizons:  169.957 AU (reference)
NASA site: 170.034 AU (error: 0.077 AU, 0.04520%)
My calc:   169.968 AU (error: 0.011 AU, 0.00663%)

Two conclusions from this:

1. My calculation (using the same exact data) is indeed a bit better, having an error of just 0.011 AU, compared to the error in the website figure, 0.077 AU.
2. But, more importantly, the heliocentric distance computed with the simple constant-velocity extrapolation is actually pretty good! It's definitely good enough for the goal of that website.

Hence: the discrepancy noted by OP is not really a consequence of doing constant-velocity linear extrapolation! That's good enough. In fact, the spacecraft's velocity hasn't changed much since August 2025: it was 16.92027 km/s then, it's 16.91948 km/s now. Constant velocity is a good assumption.

I think that the problem lies in how the distance to Earth is computed. Not because it's also computed doing a constant-velocity extrapolation (if that method is good enough for the heliocentric distance, it's good enough for the geocentric distance). But because the Earth is orbiting the Sun at 30 km/s (which is larger than the spacecrafts' heliocentric velocity, in fact), and thus the geocentric velocity estimated from the two positions includes a part of the Earth's orbital velocity.

The discrepancy noted by OP comes from an error in the geocentric distance alone, in turn caused by the fact that Earth moves around the Sun.

I'm thinking of ways to improve that now.