Does this mean the players spend all intended hit dice for healing and then benefit from song of rest? [...]
I believe you are looking at this the wrong way and, thereby, make the feature more unwieldy then it is. At the end of a short rest you can recover hit points by spending Hit Dice. I quote from Short Rest on page 186 of the PHB:
[...] A character can spend one or more Hit Dice at the end of a short rest, up to the character’s maximum number of Hit Dice [...]. The player can decide to spend an additional Hit Die after each roll. [...]
The Song of Rest specifies that (errata worked into original text in square brackets):
[...] If you or any friendly creatures [...] regain hit points at the end of
the short rest, each of those creatures regains an extra 1d6 hit points [if it spend one or more Hit Dice].
Neither of these statements specifies that you regain the hit points all together after all your rolls have finished. Look at it as deciding how much resting, eating, bandaging, etc, you want to do during that hour and then doing it. The rolls at the end are simply the mechanical rewards.
Hence, when you regain hit points from the roll of your first Hit Dice, add the benefits from the Song of Rest if they apply. The player can then decide to spend additional Hit Dice if they should need more hit points.
TL;DR: No. Yes.