The numbers guys will correct me if I'm wrong, but I calculated that dropping from a 4.75 (early in a 30 yr mortgage) to a 2.875 (on a 40 year) would lower my payments by $170/month per every $100k borrowed, plus I would save over 20% in total interest paid over the complete term of the loan. Additionally, if I take those savings and use them to make extra monthly payments to pay down the loan quicker, I would save over 60% of the interest I would have paid on the original 30 year mortgage. (My actual savings are closer to 15% making the lower payments and 50% making the extra payments, but that's because the interest and escrow from the last 19 months of "missed" payments were capitalized in the mod.)
Depending on whether or not the modification includes a deferred amount, making extra payments is either not so smart or outright foolish IMNSHO.
If there's a deferred amount, those extra payments are paying down that amount first, a long-term interest-free loan!
Even without a deferral, there is much more efficient use of the money saved than converting it back to illiquid real-estate equity (yes, I know one can get a HELOC or a new mortgage and borrow against the real estate, but it doesn't change the fact that the real estate isn't a liquid asset, and would require bank approval in order to borrow against), only to save less than 3% (which might even be tax-deductible, depending on circumstances, making the benefit even less than that of the actual interest rate).
I fully agree with you that even if there is a temporary hit to one's credit, it is worth the benefits obtained in most cases. Though I believe that if there were no late payments prior to COVID and all COVID relief and modification T&C are kept to, there might be no hit at all.