We did something similar in our 3.5E campaign in college, and it always seemed unnecessary and annoying. Ours was just a time commitment, not gold or mentors, but it was jarring to say "Okay, so I spent a month adventuring through the Underdark, solving puzzles, picking locks, and fighting all kinds of weird beasts, but what I really needed to do to get better was a week of lifting weights and parkour?"
Why should it be jarring? It's ACCURATE.
It especially makes one wonder whether, instead of adventuring, he should spend a few more days with the mentor/training montage to grind out another level or two.
The original explantion Gary provided in the DMG was essentially to say that if you want to be a very good fighter then you don't go wander in the wilderness for 3 weeks, have 3 or 4 fights, and drag back a sack of gold - what you do is to TRAIN. You spend 3 weeks in excercise, study, practice, under the tutelage of someone who already understands more than you do about what you're trying to get better at. If you're a warrior, then in the absence of live combat on a daily basis you want PRACTICE at combat on a daily basis. This applies whether you're actually trying to be a better sword-swinger, archer, spellcaster, or whatever.
The training rules then can serve a couple of purposes. For one, it provides a certain amount of mandated downtime for everyone. It generally gets the PC's back to civilization. This is then a time when the DM can sow the seeds for new adventures and allows for a change of pace from dungeoneering to city adventures. It provides a nudge away from the PC's being mere professional tomb-raiders to opportunities to experience other aspects of the DM's campaign world (if any).
Secondly, in 1E Gygax (whether intending to or not) designed a system where with one hand he handed out heaps of treasure - but with the other he systematically and unashamedly bled it away. Taxes, spell components, cost-of-living fees, etc. And of course, training costs. All these drains on the money being gained by the PC's served as motivation to continue to seek new adventures and thereby gain more treasure. More than once Gary phrased the suggested costs of something as, "It should be just slightly more than the PC can currently afford..." He wasn't trying to suggest that DMs should be a jerk to their players and suck the fun out of the game, but the exact opposite. He wanted to keep the game moving forward - even if the only reason would be to get enough to pay for "x". Of course, once "x" is paid for the player will then find that if he wants "y" he has to go get still more loot.
Thirdly, though it's not stated I believe that training time and costs serves to ground the PC's lives just a bit more in reality. It's not just party all the time, money for nothing and your chicks for free. He was trying to structure something of the Conan-esque where PC's earn AND THEN SPEND treasure, not just earn more and more and more until they're not adventurers, just rich guys with guns. You win bags of gold, spend some and find that it's not enough to buy your PC what you need and want, then go win more, spend some of that and again find you need more. Adventuring is supposed to be a job description reserved for the PC's and they seem to have been intended to find themselves trapped into continuing to adventure to support themselves in the lifestyle to which they become accustomed.
It also makes the passing of time a significant factor in how and why the PC's carry out their planning as a group. In AD&D PC's advance at different rates. They can earn xp at different rates for different activities. PC's will level up at unpredictable times and it is NOT expected that they ALWAYS do EVERYTHING together. This can come with a certain amount of frustration but it also serves to to highlight the fact that (whether you care for the analogy or not) they are NOT characters in a video game, but that the world marches on around them and while they might adventure for a living they also deal with more mundane issues of life - like trying to finish training so they can commence the next adventure. If you look at the AD&D DMG p.37-38 he's talking about the need to track time in the campaign and the example being given has PC's going in all different directions pursuing all manner of aims, but obviously they need to eventually get BACK TOGETHER. These PC's don't live narrow, computer-programming-limited lives of ENDLESS adventure. Adventuring is simply what they do to support EVERYTHING ELSE in their lives.
Of course, at this point I probably have to say what
should go without saying. Just because there's a rule on a page (or even if it's a house rule and not even written down) doesn't mean the DM should feel obligated to blindly obey it to the detriment of the players enjoyment of the game and the well-being of his ongoing campaign. If the player just wants to level up quickly and move on, and YOU as the DM would prefer to just move the game along then DO IT. Determine what you want to use training time/costs for. Make your players aware of your reasoning right from the start. By all means APPLY the rules but don't forget why you're ALL sitting at that table.