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.

In general you want to avoid extreme chain angles.

This gear calculator has a typical 3x7 setup with 50/40/30 crank and 14-28 7 speed Shimano freewheel

https://gear-calculator.com/?GR=DERS&KB=30,40,50&RZ=24,22,20,18,16,28,14&UF=2215&TF=90&SL=1.5&UN=KMH&DV=teeth

The max chain angle is set to the smallest possible .

It shows that 50/28 30/14 and 40/14 and 40/28 are all cross chained.

In general the chain wants to be straight which is achieved by using middle gears in the middle chainring, small gears in the top and large gears in the small chainring .

1x setups can be considered to leave the smallest and largest gears cross chained because there is only one chainring position.

Possibly with modern 1x setups this isn't a problem, but with a cheap 3x crankset, removing chainrings won't help anything.

Cross-chaining is a mythical beast. Yes, it's best to avoid riding significant distances with the chain at a wide angle, but you pretty much can't shift a bike without doing it occasionally, and there are advantages to being able to quickly access the full range of rear cluster gear ratios, without having to constantly fiddle with the front. Plus, it may be that the "perfect" gear ratio for a given situation may be one that provokes this dreaded beast.

Note that cross-chaining wasn't really a "thing" with old 2x5 setups, and only gained note when the rear cog count surpassed 8. Plus, with a single front you can't use the full range of gearing without crossing the chain.

Finally, note that the diagram in question isn't even suggesting cross-chaining.