How to Create Index on Long Data Type Field in MySQL

You misunderstand what the 'rows' field is. It is the number of rows that mysql estimates it shall need to read to satisfy your query. This value can be quite inaccurate. It does not mean this is the number of rows in the result - or the actual number of rows read by mysql

I don't see a lot of opportunity for improvement.

The index you added was probably a big help, because it's being used for the range matching on the WHERE clause (type => range, key => tran_date), and it's being used as a covering index (extra => using index), avoiding the need to seek into the table to fetch the row data.

But since you're using functions to construct the financial_year value for the group by, both the "using filesort" and "using temporary" can't be avoided. But, those aren't the real problem. The real problem is that you're evaluating MONTH(tran_date) 346,485 times and YEAR(tran_date) at least that many times... ~700,000 function calls in one second doesn't seem too bad.

Plan B: I am definitely not a fan of storing redundant data, and I'm dead-set against making the application responsible for maintaining it... but one option I might be tempted to try would be to create a dashboard_stats_by_financial_year table, and use after-insert/update/delete triggers on the transactions1 table to manage keeping those stats current.

That option has a cost, of course -- adding to the amount of time it takes to update/insert/delete a transaction... but, waiting > 1200 milliseconds for stats for your dashboard is a cost, too. So it may come down to whether you want to pay for it now or pay for it later.