Are multiple indexes on MYSQL table reason for slow UPDATES and INSERTS?
To answer the specific question:

No, an extra index on (entry_date) or two would not hurt performance of updating a different, title column.

Additionally:

No, the version of MySQL, 5.6, is not too ancient, even if some modern features (like window functions) are missing. You should have decent performance with decent hardware.

We can only speculate on the issue in hand but what could be wrong or explain it:

old, inefficient hardware: check the disk specifications, measure it's preformance.

fragmented index/table. Check MySQL docs and old questions/answers here of how to defragment MyISAM tables (OPTIMIZE TABLE).


Last but not least:

Your table is using MyISAM engine, which is old news. InnODB has replaced it as the default engine in MySQL years ago. There is no active development of this engine. It lacks several features compared to InnoDB (transactions, foreign key constraints, etc) and performance in most work loads. I strongly suggest you change your table (after testing of course that your apps and procedures do not break) to use InnoDB.

Importantly to your Question, InnoDB can perform SELECT and UPDATE at the same time (usually).  MyISAM completely locks the table; the update or select must finish before the select or update can even start.

When switching from MyISAM to InnoDB, be sure to adjust key_buffer_size and innodb_buffer_pool_size.