Does having a better pitching coach make you more appealing to pitching free agents? I have heard it is a tiebreaker between two teams who offer max salaries and years.

From the FAQ:

When a free agent has multiple exact offers of any value at the ML level the tiebreakers are as follows: relevant IQs of the coaching staff and ballpark. If the players previous franchise is also bidding on him, then that franchise will receive a loyalty bonus. </p> <p> When the coaching staff is used to break the tie position players will look at the hitting IQ of the hitting coach and the fielding IQ of the fielding coach. Pitchers will look at the pitching IQ of the pitching coach and the bullpen coach. Please note that an FA will sign with the team that has better coaches in hand, which may or may not be the team that ends up with the best coach and/or staff. </p> <p> In cases where the offers are still tied and the coaching staff has been factored in ballpark is used to determine which franchise the player will sign with. Pitchers always want to pitch in a pitchers park, while position players will always want to play in a hitters park. </p> <p> When a free agent has multiple exact offers of any value at the minor league level the tiebreaker is organizational depth at the player's primary position.

FWIW, every time I've ever seen the previous franchise offer max it's not just a loyalty bonus, it trumps everything else. I've not seen a previous franchise lose out. Anyone seen it happen?

I had Harry Schulte in MLB. Played 3 and half season for me. I bid the max. Still got trumped and am not sure how.
https://www.whatifsports.com/hbd/Pages/Popups/PlayerRatings.aspx?pid=7076717

Unrelated question about coaching: I see in the Manual that coaches affect players' "development," not necessarily their performance. Do we read "development" to mean only their improvement from age 18 to their prime or also as they start to reach the age when their skills slip? Also, might a guy who receives better coaching in the middle of his career be able to delay or slow his regression?

To the point... I've been skimping on ML coaching (other than the fielding coach) because by the time guys get to the ML, they're usually almost done improving. Is that a bad idea?

Just my opinion, but I suspect that IF good coaching in the majors offsets player decline, it isn't enough to warrant overspending on coaching. IMO player decline is, in my experience, largely predictable. And in the worlds I play in, the owners who win the most - not me - are usually the ones who invest in player talent and save on coaching. Draw your own conclusion.

I agree with damag and I would add that I believe training budget has a much larger affect on decline than coaching does.