Sorry, but IMHO your method is a problem waiting to happen. It is common, but it is the way underpinning is done by people who are not specialists.
Just this week, a client of mine updated me about a job where a building facade was being underpinned by your method. One contractor was hired to dig the pits. Another contractor was hired to install the concrete. No one had overall responsibility for the underpinning. The contractor wanted my client to install tiebacks in the underpinning. I told my client to stay away from the job. Luckily, he did as I suggected. The "underpinning contractor" was trying to install 17' deep concrete underpinning piers in stacked lifts of about 5' per lift. Essentially, the contractors were going to underpin the underpinning several times. The tieback anchors were going to be installed in the upper lift of underpinning. I don't know what the contractor thought was going to brace the lower sections of underpinning. Over the weekend, rain caused excavated, unshored, underpinning pits to collapse. This further undermined the foundation wall. The building facade is now significantly damaged.
There are good reasons why underpinning should be done today the same way as it's been done for many, many years. The proper method is low tech but it works. No one has found a better, surer way to do underpinning.