Sanchin originates from the Chinese kung-fu forms "San Chiem" or "San Zhan".
Example comparisons of the various Chinese and Okinawan versions of the kata, this video shows how each style practises it:
It means (in all of the names and languages) "Three battles". Possibly a shortening of "Three Battle Steps".
While there is little to no direct evidence for the following view, the shortness of the kata indicates it was more likely to have originated from a formalised type of 3 step sparring. Something that Asian arts very often make use of early in the training of a technique. As such it would not have have been practised solo, but with a partner.
In various chinese kung-fu systems the convention used within forms to indicate a grip on an opponent is an open hand. Within Okinawan karate systems, the convention is a closed hand. When the kata was brought from China to Okinawa the local convention was used and the open hands were replaced with closed.
Now the kata very commonly has both hands raised with the grip convention. It indicates that when practising with a partner that you would have a two handed grip on that partner. Effectively a clinch position.
The kata is therefore very similar to wrestling "pummelling" drills where the two attempt to gain a superior position. This also explains the use of tension and dynamic tension when practising the kata within various karate styles and the use of the Sanchin stance.
Without a partner providing resistance when practising solo, the practitioner has to provide that resistance themselves. Today, the resistance is often used to build muscle, it's an interesting and innovative way of practising.
Or, put another way. Sanchin is a form of 3 step sparring, teaching wrestling from a clinch. The end techniques differ from style to style and are probably relatively unimportant in themselves, they would likely have changed depending on what was being practised at the time, though they could also indicate good clinch ending techniques.
I highly recommend practising Sanchin with a partner. It's ... enlightening.