I think what they mean above is to use helical piles at something like 4'-0" on center to make the house "floating" and build a new foundation wall/footing under it.
What OP described sounds more like underpinning. I usually use 4'-0" long underpinning pits. You will need lateral support (something like diagonal wood or steel rakers, and probably walers) for the lateral support. Calculating the waler/raker loads is just taking lateral earth pressure + surcharge and applying as axial loads on rakers.
1. The reason the underpinning pits are 4'-0" long is based on industry experience. For a real calculation, since the bearing wall is CMU, you can calculate the CMU wall to span 4'-0" over air and be supported on both sides, most likely simply supported. Masonry does have arching action which allows this to happen.
2. The lateral support is for earth pressure and incidental loads like wind, seismic, and vibrations that could destabilize the structure.
3. If you're removing the wall in 4'-0" sections, the quality of concrete work, formwork, and rebar splices will be extremely low. Very few contractors can do it properly to make an entire foundation wall and footings. The floating idea sounds better because they won't have to do it in sections.
4. If you were thinking of something like 8'-0" or 12'-0" sections, I think it'll be quite dangerous because CMU can't be unsupported for that span.
5. If you don't want to use helical piles or there is no access for them, something like needle beams with columns might work. It just might cost way too much to be feasible, but it's an older method of underpinning. I've never actually seen it used in modern day. Underpinning is more common in that case.
6. Make sure there is a good bond between the upper structure and the new foundations. I think some people use non-shrink grout, but I prefer to pack it with repair mortar. You might get some pushback. Also consider hammering in steel wedges every 2'-0" or 3'-0" to ensure that the new foundations are really taking the required loading.
7. If the building gets supported on helical piles, their expected settlement is quite low. In this case, your footing at the bottom will not be doing much. So you might have some leeway in designing those footings. I still wouldn't just go footing-less, just in case.
8. With helical piles, some do use a combination of end bearing on the helices and also skin friction. So make sure that it's actually designed for the skin friction it will be receiving, i.e. starting below your new foundations.
9. I might be way off about the helical pile thing. Maybe the ones above are talking about a different system.