As this is the Camcorder Section, I wonder that we shouldn't reflect on why cinema shots were considered to have shallow depth of field.
Personal view:
IN the early days of silver-based filming (which probably influence us even now), film stock was very slow - this meant that large aperture lenses were needed for indoor (ie Stage) filming. The consequence of this was that the lens was left fully open, hence the shallow DOF, since more light was a downer, needing more wires, generators and stands littering the floor.
It was discovered that shallow DoF helped the "creative" aspect of the movie - and so it become normal to add neutral density filters to outdoor filming, so the lenses can be used wide-open.
SLR's
Before digital (remember?), we had optical viewfinders and wide-open focussing - so most cameras came with a wide-aperture prime lens so the camera was easy to focus. More recently we had AF and most zooms were modest apertures (Cost!) so the viewfinder image was dimmer, but AF saved the day.....but in general SLR (and DSLR) lenses will offer a reasonable aperture to make focussing/composition easier.
Now:
Whilst I read and hear folks saying was has been said earlier in this thread....that DSLR produce more-professional-looking videos - there are some folks that will disagree. A good quality (ie Pro-sumer) camcorder will beat a DSLR, - my reasoning is this...they are easier to handle, being arranged for the purpose. They offer focus and exposure features to help the photographer and they have a ability to change gain, as well as offering very acceptable on-camera Audio recording, perhaps using an external XLR pair of mics. Very few of these features are available in DSLRs, mainly because the manufacturers still think their customers are taking Silent Stills.
Professional films
We've seen mobile-phones used for filmmaking - and they can be good in situations where poorly exposed/focussed footage helps the storyline....but I doubt anyone can point to any serious technical advantage, other than they can be destroyed (being cheap) . .. indeed sometimes this applies to DSLRs used in film production (eg 1-off Stunts - who wouldn't strap "just one-more" DSLR to the crash-scene, to be sure?), esp. knowing that a large frame video will have plenty of definition to fool most audiences, after some processing in Post.
Back in Post#2 I questioned if OP had the necessary skills to produce/edit their Promo vids - and (for me), that is still an issue, whatever camera is chosen . . . . but I think a camcorder at that budget probably comes out on top UNLESS the cameraman is attempting something experimantal - and has the necessary skill to know how to achieve the effect . . . . the quality of footage from say a Z1 will nearly always beat a DSLR, despite it being "only" a camcorder. Mindyou one can still tire of it, as they weigh quite a bit compared with compact and APS-C non-mirror DSLR.
Stated Budgets
OP suggested "under £2k" so I wonder if Posters might give their choice(s) at this level ? . . . may I suggest as a second choice we can vote for a cheaper solution that will include a second camera, within that budget. Bear in mind that ideally the cameras are "the same" to make scene-matching easier.