I'm trying to understand what is the faster way to get a strong sentry to the front at the beginning of the round (the setup time).
Is it better to build a sentry directly on the front? This has the advantage of it being able to shoot enemies even before it's fully upgraded, but the disadvantage of the engineer having to scavenge metal around the front to build it.
Or perhaps it is better to build a sentry near the resupply, then haul it? This has the advantage of metal being readily available near the resupply, but the disadvantage of having to haul it (slower move speed) the whole distance, and also that it will not be there at all until the engineer arrives. Also, the hauling method incurs the overhead of deployment time twice.
Best Answer
This is going to depend on the map, how far the resupply cabinet is from a building area, where you want to deploy your sentry, and what sort of ammo supplies are near the desired deployment area.
If you know there is a full ammo pack, or a couple of a 100-metal ammo packs, near your desired location, you should probably just build it there and get resupplied without going back to spawn. You will fare better if you drop your dispenser before your sentry as well, and make sure to help it build faster while you wait for ammo to respawn... this will give you an extra ammo boost for upgrading your sentry. In general I think this is the faster option.
However, if you know of a good sentry position that has no ammo nearby (or only a small 41-metal item), and you won't have too far to run between the resupply cabinet and a buildable area, and that buildable area is pretty close to the place where you want your sentry, then this option can be faster. I'd say for most maps it isn't the better option. It does have one nice feature though: you can keep the sentry out of danger until the enemy uses their first ubercharges and avoid rebuilding the gun.
Above all keep in mind that it's almost always better for your team to build a level 3 teleporter and dispenser than it is to have a level 3 sentry. A level 1 sentry with player support is still pretty effective, and the healing of your dispenser and fast replacement of allied players is MUCH better than the extra damage you get from a level 3 sentry in almost all scenarios. Also, offensive teams almost always build an ubercharge or two during setup time, which they will often use to immediately clear out any sentries that are nearby (but will rarely use to target teleporters and dispensers).