Your question is based on a faulty assumption:
Since debt-to-income ratio (dti) is an important factor in determining one's creditworthiness especially for mortgage rates,...
Debt-to-income ratio is not directly related to one's credit worthiness, and also is not relevant for your credit score. For example, even someone with no income at all could still have a very high credit score.
In general, your credit score helps determine your mortgage interest rate (the better your score, the lower your rate). Once your rate is determined based on your score, the DTI is used to calculate the maximum amount of mortgage you can afford.
Most mortgage rates bottom out when your score is in the range of 740-780. (Once your score is higher than that you won't get a lower rate.) If your score is below that range and consequently you won't receive the best interest rate, you may be able to make some minor tweaks to your spending habits to bump your score into the better tier. Note you only need to do this 1 month before you apply for the loan, and only need to continue doing it until the loan closes.
Regarding the situation you described of spending $2K from your debit card instead of your credit card, this might be helpful if you are not in the best tier of interest rates AND if your total revolving credit is low AND if your CC is reporting the $2K as your current balance. But the reason it helps is not due to DTI, but instead utilization percentage, which makes up 30% of your credit score calcuation. If your total overall revolving credit is $4K, and at the moment your CCs report you have $2K in utilization, your percentage will be 50%. Going from 50% to 0% will certainly bump up your score, and possibly put you into a better tier for mortgage interest rates during the month you do this. But if your revolving limit is $20K, your utilization percentage would only be 10%, and knocking that down to 0% probably won't bump your score very much.
As for DTI, the difference between $2K and $0 on a CC probably only affects your minimum payment by about $50-100, so that is likely negligible in changing how much mortgage you can afford (say $10K in additional mortgage, and if you're already that close to your affordable limit you are likely buying too much house).
All that being said, you did manage to back into a mostly correct conclusion:
...it is clear that one should keep the monthly recurring debt and credit card payments as low as possible.
This might help you for the reasons listed above, but you don't need to do this on a regular monthly basis, only about 1 month before you apply for the loan, and then only until the loan closes and the funds are disbursed.
Tip: instead of using your debit card or cash, you could also find out what date your CC reports on, and just make your pay-in-full payment a few days before that, and then you essentially accomplish the same thing but still reap the benefits of whatever CC points/cashback you may have.