Repair chimney on the cheap

2 ways to avoid reconstruction

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Q: I have a 50-year-old one-story freestanding brick chimney with one flue (see photo).

A chimney sweep recently did an inspection and noted the part of the chimney extending above the roofline could be tilted away from the structure by pushing on it. Apparently, the bond at or near the roof flashing had been compromised perhaps 10 years earlier when a new roof was installed.

The sweep recommended the chimney extension be reinforced by removing the mortar cap, inserting rebar into the four corners of the chimney down to a sill, then filling the void with mortar/concrete.

Is this method of construction considered appropriate for the aforementioned problem or should other methods be considered?