actually this is how you do a tempo!
I would rather do it in my racing flats - but I train in racing shoes - so that is just me, but the idea that you have to get warmed up, etc for tempo is wrong headed thinking
a tempo is not a race -
the idea is to flow into a tempo
find the pace over time - not jump right in and try to hit some random goal time and hold it for some random time or distance
your coach, in my opinion, knows what he/she is doing and you should stop thinking there is some better way to do these workouts.
I ran sub 24:008k mostly off of these kind of tempos - where I just cruised into them - no stopping, no pre-stretching...the warm up is the first few miles just as your coach is having you do...running by feel is exactly what you should be doing...I timed mine but that can be a big crutch if you become a "slave to the watch" = meaning become too concerned with what splits you are hitting and such.
the best tempo is where you are constantly monitoring your efforts - so at times you are holding back and at other times pushing it a little -----not where you are grinding it out the whole way - the goal is not to see how fast you can run - it is simply to find a pace/effort that you can hold without allowing your breathing to get out of control...a pace you could hold for another several miles - so that when you finish you are not exhausted...
listen to this coach - they know what they are doing!