This was never directly addressed in the show, the novelization of the pilot, or the tie-in comic book. So we'll have to examine the possibilities.
The McKay/Carter Intergalactic Gate Bridge was constructed by the following:
Place gates at strategic intervals
Thirty-four Gates from both the Milky Way and Pegasus Gate systems have been strategically placed in the massive void between our two galaxies.
There are several important points about these 34 gates:
- They were under the control of the SGC (i.e. no enemies were using them)
- They were known to be operational
- The locations (gate addresses?) were known
None of this is true for Destiny. If any enemies have taken control of gates, or any have been destroyed (or gotten too close to a black hole), then they aren't any good for Destiny. And do you know the gate addresses for all of them? It's possible that they are maintained by Destiny and the ship can update for stellar drift, but once you start your journey, you won't be able to refer to that.
Write macros to be able to forward someone being transferred
A macro that I have written specially for the occasion will command each Gate in the chain to store you in its buffer and forward you along to the next, and the next, and the next, and the next, and the next until you arrive here.
[...] enter the Milky Way Gate system where a similar macro designed by yours truly will forward you along to the SGC
McKay wrote two distinct macros: one for the Pegasus Galaxy and one for the Milky Way Galaxy. We don't learn much about what this entailed, but this suggests there isn't a generic macro that can be used for any arbitrary galaxy's gate system.
If I were to guess, the macro involves specific gate addresses for each gate you are forwarding to, so you would need to take the time to understand the gate system of each galaxy and know which gates are viable. Or if gate addresses don't work between galaxies, then it involves a custom gate coordinate system.
Have a space station to transfer between gate networks
Once at the midway space station you simply exit the Pegasus Gate system and enter the Milky Way Gate system
Each galaxy's gate system can't interface with each other, so you need a place to stop between systems. Destiny doesn't have Puddlejumper-sized craft that can fit through the gate, so you need somewhere to stand.
Theoretically, you could use a ZPM to go from one land-based gate to another land-based gate in a nearby galaxy in order to prevent needing a stopover point, as the SGC has used to go to the Pegasus and Ida (i.e. Asgard) galaxies. But since Stargates aren't usually designed to connect to ZPMs, you'd have to macgyver a way to connect each Stargate to a ZPM, and have more than one available so that you could leapfrog them through the galaxy. Given the difficulty that the Atlantis expedition had in finding even one ZPM, and there being doubts about if Destiny even has the technology for a ZPM, this is unlikely to be a solution.