He told BBC Scotland: "I am a fairly pragmatic person, but I would be an odd sort of character if I did not feel a degree of anger.
"When I think of all the work we have put into creating a cogent, coherent strategic plan, it is obviously frustrating that I will not be there to finish the changes that are needed to take Scottish Rugby into the 21st Century."
Clubs had been angered by suggestions they should lose some of their policy-making power under the SRU re-development plan.
Mackay added: "I am very strong in my view that we cannot have a strategic plan without proper governance.
"Some on the committee feel very differently from that, but they appoint the chairman and that is why I am no longer chairman.
"It is a business as well as a sport and any business needs a structure that can take rapid decisions. All other unions have it."