The reason why #2 doesn't have the same effect as #1 is
scale.
Wealthy people are not affected by prices in the same way as average people.
I hypothesize that being able to buy more with the same income rather than being given a higher income helps the wealthy more. Let's take a practical example:
A banana now costs 50 cents. It would have to cost $2 if we paid for higher labor costs, for that $20+ minimum wage.
But if you're upper-middle class or above, or way above, it doesn't really matter if that banana costs 50 cents or $2. Heck, for some people, it wouldn't really matter if it cost $50 or $100 or even $1,000 if you're Jeff Bezos.
There are only so many bananas you could possibly want.
In this way, I think that option 1 would be more beneficial for the common person. Sure, they'd have to pay for a $2 banana, but they'd be making a more comfortable salary and they still only want one banana. Meanwhile, Moneybags Factory Owner would have to dispense their wealth to labor instead of hoarding it in capital.
I'd also argue that we don't do #2 "because of competition," instead that it was an intentional choice not to tie minimum wage to inflation and to keep it at rock-bottom levels. That is corporations-write-the-laws policy. If the minimum wage were raised there would absolutely still be competition and companies would still be trying to make productivity gains from option #1.