Appleton was forced to part ways with his entire first-team squad at the start of the season to ensure that the club was not liquidated, but he led Portsmouth to 16th place in League One before his departure to Blackpool earlier this week.
Birch told BBC Radio Solent: "There has been a lot of interest in Michael over the past few weeks with the various managerial vacancies.
"It is not altogether surprising that he has left but still disappointing and just another hurdle for us to get over.
"It was very difficult for Michael and not the job he signed up for so you have to have some sympathy there. In very difficult circumstances, he did a good job."