I think this is a very interesting thread so well done in raising this topic however I disagree with your conclusions.
Doctor
Doctors don't do much but they are like insurance you hope to never use them but when you do you are happy. For a new team I think the critical thing is to get three trainees in place and try to promote from the bottom division in their first season even if it means geting relegated the season after. They will still get the promotion money, have had much larger revenue and then will potentially get a $1,000,000 draft pick. To do that they need their players functioning well. For that they need good game shape. New players that understand game shape and enthusiasm will wipe the floor of other new managers with much more expensive rosters. This is where a doctor comes in. One with massage speciality will help your players stay in top shape and that will give you more wins. Ever lost a game by a few points? A massage doctor could have changed that loss into a win.
So for any new team I would suggest getting a level two doctor with massage speciality. $5,000 or less a week.
PR
I doubt a new team is going to get good value out of the PR guy getting an increace in merchandising but again I think the specialities are gold. They should look at their schedule and see do they need help with home games or away games and get a CI or NA PR guy. I like winning games and I assume new players do too.
So for any new team I would suggest getting a level two PR with either national appeal or crowd involvement speciality. $5,000 or less a week.
Trainer
Training has been greatly devauled in the game by the crazy drop in player values but it is still one of the most fun parts of the game. It also binds players to the game when they have a particular player they identify with. There is little reason not to train unless you are trying to run a 5-7 man roster and just training game shape each week, New players usually don't do such pathetically unrealistic things (although the game rewards those that do but that is a different issue) so they should train. At the moment you can pick up a decent 19 year old with high potential or an 18 year old with good skills but low potential for basically nothing. Add in a level four trainer and you will be a very effective trainer and have a roster that by the end of the season is playing like players with a much higher salary.
Even better again there is the speciality. A lot of new teams buy old players. Skill drops are very painful. Why not reduce them with a career extention trainer. I don't like stamina drops so fitness is also great.
So for any new team I would suggest getting a level four trainer with either speciality. $10,000 or less a week.
In total this is $20,000 a week which is not peanuts for a new team but it will equal more wins and wins are what new teams in the bottom division should be about. Add in some improved game shape, less of a penalty for away games, less skill drops on your old players and the training on your young players and your team must be at least 10+ points better a game by the end of the season. Well worth it.