Did you try the following?
just append time in seconds #t={seconds} to source URL:
<video controls width="360">
<source src="https://test-videos.co.uk/vids/bigbuckbunny/mp4/h264/1080/Big_Buck_Bunny_1080_10s_1MB.mp4#t=0.1" type="video/mp4" />
</video>
I have chosen a fraction of second (0.1) to keep number of frames small, because I have the suspect that if you put 1 second, it would “preload” the first 1 second of video (i.e. 24 frames or more ….). Just in case …
Works fine on Chrome and Firefox on desktop 🙂
Works not on Android mobile, though 🙁
I did not test on iOS, iPhone, IE yet ??
Edit May 2021:
I realized that many modern browsers now show automatically a poster of first frame.
Seems like they heard us 🙂