I still don’t get how if I start trying a proof of work, and someone faster than me finishes the same proof of Work first, I would never mine a single bitcoin? Is that even the right question?
It does happen that two blocks are found at nearly the same time, before adjustments can be made. In those cases one of the two ends up getting 'orphaned', it's relatively rare compared to normal blocks though.
That's right.
Now, probabilistically, you'd occasionally get lucky and get the full (big) mining reward, in proportion to your fraction of total hash power.
The alternative is that you join a mining pool, and then receive the same amount (in expectation), minus the pool fee, but with much smaller variance, as the larger pool will much more frequently get lucky and then distribute the reward (minus the pool fee) to the contributors in proportion to their hash power.
But either way, your expected reward will be approximately the same, whether or not you work in a pool (modulo the pool fee), it's just whether you get a large amount very rarely or a small amount very often.