linguistics-style trees in Metapost

Disclaimer: this is in Metapost but it's not really a proper answer to the question.

It is, of course, possible to draw anything nicely with Metapost, such as this version of your tree:

but as you can see below, I did not do that using a qtree-like syntax. Whether you think this is "easy" is a matter of opinion, experience, number of trees you want to do, etc.

You could invest time in writing some MP that processed qtree notation, or in patching metaobj, or in writing macros for your own notation, or you could just draw trees one at a time, like this:

prologues := 3;
outputtemplate := "%j%c.eps";

beginfig(1);

    numeric dx, dy;
    dx = 34; dy = -13; 

    z0 = origin;
    z1 = z0 + (-dx, dy);
    z2 = z0 + (+dx, dy);

    z21 =  z2 + 3/4(-dx, dy);
    z22 =  z2 + 3/4(+dx, dy);

    z100 = (x1, 3dy); 
    z200 = (x22, 3dy);

    draw z1 -- z0 -- z2;
    draw z21 -- z2 -- z22;

    label.top(btex VoiceP etex, z0);

    picture p[];
    p1  = thelabel.bot(btex DP etex, z1);
    p21 = thelabel.bot(btex Voice etex, z21);
    p22 = thelabel.bot(btex {it v/}P etex, z22);
    p100 = thelabel.bot(btex strut He etex, z100);
    p200 = thelabel.bot(btex strut runs etex, z200);
    forsuffixes $=1, 21, 22:
        draw p$;
    endfor
    forsuffixes $=100, 200: 
        draw p$ withcolor 2/3 blue;
    endfor
    
    vardef connect(suffix a, b) = 
        draw point 1/2 of bbox p.a -- point 5/2 of bbox p.b dashed withdots scaled 1/4;
    enddef;
    connect(1, 100);
    connect(22, 200);

    undraw bbox currentpicture; % add a small margin
    
endfig;
end.

On the specific issues you have with metaobj:

• I cannot see any simple way to get it to draw the lines as you want -- it is aiming at the centre of each node and cutting the lines at the bounding boxes, this will mean that the lines are usually at different angles. There does not seem to be any option available to make it draw lines at regular angles.

• Centering multiline labels is a plain TeX issue, rather than a metaobj problem. Each label goes in an hbox as usual, so you could add a block like this near the top of your file

   verbatimtex
   defs#1{vbox{let\crhalign{hfil ##hfilcr #1crcr}}}.  
   etex
  

  then you can do things like: label(btex s{Above\below} etex, origin);

Disclaimer: It is not a solution in Metapost, but it could be useful.

As Alan Munn commented, TikZ/PGF is generic, so tikz-qtree and should be portable to Plain TeX and ConTeXt. forest is also based on TikZ/PGF, but it is tricky due to its LaTeX dependencies. The following is an example of tikz-qtree in ConTeXt I'm not proud of, but it is a proof of concept.

usemodule[tikz]

unprotect
%Shamelessly taken from:
%https://wiki.contextgarden.net/LaTeX_Math_in_ConTeXt
let@@end=end
appendtoks
letend=@@end
to everystoptext

%This is really ad hoc.
%It will possibly break things when you try
%loading other packages better suited
%to LaTeX
defbegin#1{%
    csname #1endcsname}
defend#1{%
    csname end#1endcsname}

input pgfsubpic.tex
input pgftree.tex
input tikz-qtree.tex
protect  

starttext
starttikzpicture
[level distance=40pt]
Tree [.S [.NP [.N this ] ]
[.VP [.V is ]
[.NP edge[roof]; {a noun phrase} ] ] ]
stoptikzpicture
stoptext

For some reason, a similar approach for Plain doesn't work for me and I didn't expected as I'm not a Plain user, but is feasible as you can see.