Why doesn't InnoDB store the row count?
For starter there is no such thing as the 'current count' to store in a variable. A query like SELECT COUNT(*) FROM ... is subject to the current isolation level and all concurrent pending transactions. Depending on the isolation level, the query can see or not see rows inserted or deleted by pending uncommitted transactions. The only way to answer is to count the rows that are visible to the current transaction.

Note that I did not even touch the even more thorny subject of concurrent transactions that start or end during the count. Not to mention rollbacks...