Correct way to shift?
Is it not literally suggesting cross-chaining?

Cross-chaining is putting a chain at extreme angle between front and rear cogs. The picture above illustrates correct ways to use the front triple. Incorrect ways would be: the biggest front ring with the biggest rear one, and the smallest front one with the smallest rear one.

I'm also thinking about removing two of the chainrings since only one chainring makes more sense.

I've seen/used the Tourney group and agree that the ratio range it provides is unnecessarily big for flat road applications. However, removing all but one front chainring, in my opinion, will leave you with a very narrow range at the back. Most people will be comfortable with existing two front chainrings of the Tourney setup (the smallest and the middle one), as the biggest front cog only makes sense for quite steep descents and high speeds, unusual for leisure cycling (or at least for non-mountainous terrain).
Note that chainrings on Tourney are riveted not screwed to the spider, so removing them will require some drilling or smashing. Putting them back on will become problematic as well. That is why I do not recommend it.

Say, I make it a 1x setup after removing two chainrings, (keeping just the middle one), will I still meet with cross chaining?

Strictly speaking, yes. The 7-speed chains were not designed to bend through the whole range of the rear cassette. Having a middle front cog with either the biggest or the smallest rear cog will constitute cross-chaining, albeit less grave. It will most likely work, just that it will not be in the intended use/range of the setup, and might lead to decrease in chain/drivetrain life, decrease in shifting reliability, may produce additional noise etc.
On top of that, you may start experiencing chain drops if you remove the front derailleur cage, serving as a chain catcher. True 1× setups have additional measures to increase chain retention, which Tourney does not provide. Again, in the Tourney case, the gearing range will just be very limited.