Take a look at this link here : https://www.historicaltuning.com/Harmonics.html
It gives an example of particular keys on the keyboard, and discusses harmonics. One of the G keys shown happens to be the third harmonic relative to a 'fundamental' frequency note that happens to be one of the 'C' keys. And there are information sources out there (such as physics and applied science) that discuss adding particular harmonics together -- eg. if you add the 'fundamental' (aka first harmonic) and the third harmonic together. When they do it, some sort of 'constructive' (non-clashing) effect occurs when we look at the resulting 'summed' waveform (combined result). But when we focus on the third harmonic ----- this one, and the first harmonic ----- the transition from third to first harmonic (aka fundamental) appears very natural - without clashing ----- due to their particular constructive, non-clashing nature.
Also, when we're talking about a harmonic, it refers to a sinusoidal behaviour. And a sinusoid can be defined by a frequency, and an amplitude, and also a reference point in time. So if we compare a 'fundamental' sinusoid with a third-harmonic sinusoid, then we would be able to see that the peaks of both the fundamental and the third harmonic sinusoid are "symmetrical" (in a time plot) ----- symmetry around the point in time where the fundamental has a peak (highest point). Eg. see this link here : https://electrical-engineering-portal.com/harmonics-what-are-they-what-do-they-do
However - unlike music notes, the information in the above link does NOT have the third harmonic being at the same 'amplitude' as the fundamental. But that's ok, as that information is under a slightly different context. We're just borrowing their diagram only.
And the third harmonic just so-happens to have its highest point there too - at the same time. So this relates to the strongly additive nature of fundamental components and third harmonic components. They're sort of (or strongly) on the 'same page' ---- or a nice match-up. So hopefully these details can help go toward getting ideas about why 'dominant' notes transition nicely (resolve) to fundamental/root/tonic notes.
Although ----- when we strike a particular note on the piano, such as hit a 'G' key, then the sinusoidal waveform won't necessarily be (at first) lined up (in time) with the 'fundamental' waveform. But - maybe - as seen in some science experiments, some physics mechanism occurs, in which the waveforms become aligned. And maybe electronic tone generators may need to have this sort of alignment as well. Or --- maybe human ears aren't sensitive to timing differences when we transition from a third harmonic note to a fundamental note. I thought I'd just mention it - just in case!
Edit (addition) - a contributor in the comments section made an excellent point - in that - for example, when G resolves to C, the two notes are not sounded simultaneously. There is certainly an interesting link between third harmonic frequency and fundamental frequency.
Just giving an example - to be clear about third harmonic frequencies. Take the dominant chord (G-B-D) from C major. Then consider the root chord of C major, which is C-E-G. In the frequency chart --- you'll find that G (ie. first note of the dominant chord) is the third harmonic of C (ie. first note of the root chord). And B (2nd note of the dominant chord) is the third harmonic of E (2nd note of the root chord). And D (3rd note of dominant chord) is the third harmonic of G (3rd note of the root chord). So - when comparing counterpart individual notes between the dominant chord and the root chord --- we have three 3rd harmonic notes transitioning to three fundamental notes (when we transition from a dominant chord to a root chord). Quite interesting.
Just to be extra clear ----- let's take the note C2 on the keyboard, which has a frequency of 65.406 Hz. The third harmonic frequency is three times that frequency, which is 196.218 Hz, which is close enough to 196 Hz, which is the G3 key. So that takes the C to G transition into account.
And let's take the highest note of the root chord, which is G2 (97.999 Hz). Then the third harmonic frequency will be three times that ---- ie. 293.997 Hz, which is close enough to D4 --- which happens to be the highest note of the dominant chord (ie. highest note of G-B-D is 'D').
And one extra observation that could be important too is ------- note that the highest note in the dominant chord (for our example) is a 'D'. So - when we play the notes of the dominant chord one note at a time, we get G, then B, then D. And if we immediately follow on by playing all three notes of the root chord (all at once ----- C-E-G simultaneously) ....... then the overall result - to our ears, will actually sound like playing these four notes (individually) in sequence ---- G-B-D-C. The step-down from D to C sounds natural ---- a stepping down of the 'major 2nd' from D to C. And - the 'C' note (being the root/tonic note of the root chord) will have its side-kicks E and G (as part the root chord). And those two side-kick notes will just be secondary ------ as 'C' will have the biggest influence in the root chord. That is, even though the root chord is played (with C, E and G played simultaneously), it really sounds more like an overall 'C'. And we just feel that this sequence --- G-B-D-C ----- like a 'done deal' when we play it. Or, alternatively, if we play the dominant chord notes (simultaneously G-B-D), followed by playing the root chord (simultaneously), then --- to our ears, it just sounds overall like playing a 'D' followed by a 'C', which also sounds like a 'done deal'. This sort of natural terminating pattern certainly occurs when the two chords of interest happen to be the dominant chord and the root chord.
Also - there is also the situation about what happens in our mind (or brains) - in the way we process sound information. For example - somebody or some animal that hasn't been conditioned to recognising and accepting audible frequency sound sequences - such as a transition from a 'dominant' to root - probably wouldn't make much of this interesting pattern. Maybe getting into realm of 'psycho-acoustics'.