Masonry is hard to work with especially moving walls or expanding -- ergo expensive.Today, there are issues with re-enforcement for earthquakes and permitting, so be careful.

I am a 18yr+ construction Tradesman with multiple professional trade level experiences Carpentry (Framing/Finish), Painting, Flooring, Total Drywall, Masonry, Concrete, Electrical & Plumbing. excluding HVAC & Carpet.. the other trades I get it done correctly w/o having to redo work, I like to do my Projects on time..

The veneer is attached 1 of 2 ways... 1)  mortared right to the board with no air space or 2) the use of masonry anchors at 16" on center (horiz and vert) with an air space and possible rigid insulation against the board. 

we are in house on windows, roofs, and masonry..... we still sub out the work (except masonry) but we are able to save a little and also work with other investors for other streams of income

Rule of thumb masonry (cmu & brick) dimensioning is:  Even feet the inches will be 0 or 8 for full brick coursing. 

Don't paint masonry.

That said, as @Pat L. mentioned, running stand alone flue pipe through a masonry chimney is not uncommon at all, for the very reasons you're asking.Keep in mind though, you'll likely need a custom cap made to keep out the elements and critters.On another note, if the chimney is "flaking all over the place", is it sound?

(Danny Wells Masonry, Redemption Roofing, Wood-n-Deed) We also had to hire a lawyer when Beam refused to refund any of our money or fix their damage.

A small masonry chimney was pulling away from a stucco house.

You might have to dig down and add on to the footings.ETA: Patricia has a good point about contractors not wanting to deal with renovating masonry houses.