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I disagree. Look at 'simple' problems like Fermat's last theorem.

It took mathematics much beyond what was known at the time to get anywhere close to a solution.

And much beyond what 99.9% of people will ever hear about in their lifetime.

> I'm saying one can make some progress without much theory.

Claimed without evidence.

Sure, you can make trivial progress by writing a computer program to search, and the problem is "elementary", but "elementary" doesn't mean "can be solved without studying lots of techniques and theories", it means "can be solved without calculus"

A good example of this is Marjorie Rice, an amateur mathematician, who discovered four new types of tessellating polygons.

[0] https://mathtourist.blogspot.com.au/2010/06/tiling-with-pent...