With regard to the use of CCRs on Part 90 frequencies, the FCC
has done something. In September 2018, they issued this
memo regarding the use of radios in Part 90 which can be programmed from the front panel (which includes many of the CCRs). Additionally, many CCRs have received FCC equipment authorization grants (aka, type acceptance), but that doesn't mean that the radios are being used in accordance with those grants. For example, some CCRs ship with front panel programming (FPP) disabled in order to comply with FCC rules, but the procedures for unlocking the FPP are widely known. Another example, some CCRs have FCC equipment authorizations for Part 90, but with oddball emission designators (bandwidth) that nobody actually uses or with much lower power than the 5 watts that everybody uses.
With regard to the use of CCRs on Part 97 (amateur radio) frequencies, those of us with Part 97 licenses are expected to know the rules and abide by them. Unfortunately, many amateur radio operators are not up to speed with the provisions of §97.307 of the amateur radio rules, specifically paragraph (e) which covers spurious emissions from transmitters operating between 30 and 225 MHz. I have tested many handheld radios at hamfests using the same procedure that the ARRL Labs uses. Just about every Baofeng I've tested exceeds the spurious emission limits in §97.307(e). I own one Baofeng UV5R that was sent to me about a year ago and it's clean with regard to spurious emissions. Every Icom, Kenwood, or Yaesu handheld I've tested is also clean with regard to spurious emissions.
Inexpensive Chinese radios has been cussed and discussed in these forums, and others, ever since the first cheap Wouxun radios appeared at the Dayton Hamvention several years ago. The positives and negatives of these radios have been covered many times. I believe that the OP has the answers to his question, so we're done here.