For example, could the sentence ”我懂四川“ sound natural at all? If so, could it mean you know the place very well, possibly even in a nostalgic or essential sense?
了解 will be more appropriate for the sense of
know the place very well
我懂數學will be a different sense.
I am good at mathematics
If Octopodes is looking for a recent work, from a short paragraph of the following author (in 2022 Apr)
我懂 (the title)
文 / 宣教部 蕭錦鋒 主任牧師
I had experienced (for this short paragraph)
”我懂四川“ - I understand Sichuan, is an objectionable claim without followed by a specific subject such as 四川話, or 四川人. Otherwise, it equates to saying "I understand New York, which is a pointless sentence.
In this case, 了/瞭解 is the correct term to use, which has the same meaning as 懂, but with a more comprehensive and non-subject specific sense - 我瞭解四川 = I understand Sichuan well (around).
As a Chinese, I think we often use sentences like "我了解四川" instead. "我懂四川" often appears in literature, with the effect of personification, just like SiChuan is a person and I understand her very well. You may briefly translate "我懂四川" the same as "我了解四川", but actually the former sometimes implies a stronger emotion than the latter. Anyway, it is necessary to decide its meaning through the context.