Use FLCH? How does that differ from using VNAV or v/s to climb 1000'? Or differ from VNAV or v/s when trying to manage climb rate to avoid the level off/climb/level off/climb aspect of some departures(and arrivals)?
LVLCHG differs not in climb from VNAV. However it does from V/S. So much as to that it allows you to climb with full power.
Thus there is a danger when using V/S inappropriately at higher flight levels, such as cruise climbs. As using V/S has its design emphasis on achieving the rate of climb allowing low speed to develop until low speed protection activates as it pitches for a selected rate and then attempts to achieve the speed selected while maintaining the selected rate of climb. This could lead to an under-speed situation to occur if not used with caution.
However, equally using V/S has its benefits for the same cruise climb if aircraft far exceeds the maximum 1000'/min due performance capabilities.
IMHO there is thus a time and place to use either/or other, neither is exclusively correct. Good airmanship will dictate which option to use.
Back to the OP...
Departures with new climb instructions, when given in UK airspace, should include the term NOW or UNRESTRICTED if the controller would like you to climb without adhering to the published altitude restrictions on the SID. Thus if cleared to climb, VNAV will level off at intermediate restriction unless you remove it, this is safe. The closure rate of VNAV is commanded by the ACQUIRE manoeuvre of your jet, thus may exceed 1000'/min, which should therefore be actioned by yourself if you observe this to happen using V/S. In all other cases when climbing from e.g. 4000' to FL120, this should not be done using V/S, rather use full/derated climb power, as ATC monitors your rate of climb and will give further clearance or handover when approaching your assigned altitude/level. Reducing the climb rate using V/S will give ATC possibly the idea that you are limited in climb capability performance. Further climb clearances to new altitudes/levels may take this personally introduced limitation into account without your intention.