This probably isn't as much of a problem as it appears to be, since the content you can do is gated by quest progress, and quests also give you gear appropriate to their level. If you're level 40 but only doing level 25 content, it's perfectly fine to have level 20-25 gear since your level will be synced down in group content.
I must admit it's been a while (3 years) since I started playing myself, but I don't recall there being that much traveling around before getting my own mount. You stay at one quest hub for about five levels before moving on to the next. I suppose the visit to the Scions' headquarters in Western Thanalan and then being sent to Eastern Thanalan pretty much straight away is quite a lot of travel in a short amount of time. The chocobo mount quest happens after that part and is a level 20 quest. I agree that the chocobo porter network is weirdly fragmented though.
Be sure to also make use of other cheaper forms of automatic travel besides the chocobo porters. The Return ability gives you a free teleport every 15 minutes to a destination you've set beforehand. You can set three favorite destinations for teleports at half price, and if you use an authenticator to log in you also get one free teleport destination. Airship travel between the major cities costs 120 gil, which is quite a bit cheaper than teleporting.
I haven't used HoH for leveling myself so I don't know how efficient it is compared to other methods, especially with the waiting times. But the various duty finder roulettes also give good exp, particularly leveling and alliance raids (and MSQ if you have the patience to sit through the cutscenes). Then there's FATEs, beast tribe dailies and hunts, which you can do while waiting for duty finder to pop. Personally I prefer to do a variety of things rather than single-mindedly grind the most efficient method. Check out the challenge log as well, some of those give good chunks of exp.