"The problem is that....". Good or bad English?

"The main point is that ...", "The problem is that ..." are perfectly fine. The alternatives you mentioned are inferior, if not incorrect. The Corpus of Contemporary American English returns 2391 results for "the problem is that", while it only gives 124 results for "as the problem", none of which are used in the style that you (or to be exact, the reviewer) suggested.

Reference:

Corpus of Contemporary American English

Agree with Masoud-Ata that your constructions are fine.

Some times (many times?) the word "that" can be left out of a sentence with no harm. Sometimes, "that" is unnecessary and leaving it out can actually make a sentence cleaner. If you have strict word counts for a paper, eliminating "that"s can help stay within limits. And sometimes "that" is overused and causes the writing to sound repetitive.

But just because "that" is often optional does not mean that it is always prohibited. Some people are given the advice of avoiding "that" in school, and they take it as a hard-and-fast rule rather than the stylistic advice that it really is. I suspect your reviewer may be one of these overly-concrete people who demand that if "that" could be taken out, then it must be taken out. (I've worked with one or two myself!)