Not only is LVH fine, it's the only option.
This is categorically untrue.
Only light vehicles can carry the LOG module. (Infantry get the exclusive LOG-S module which is less tonnage-efficient but slightly cheaper per unit of supplies and, as you say, has to be directly in the formation where resupply is needed. Critical for boarding teams and potentially good for garrison forces.)
You don't need very many INF+LOG-S in a formation in practice. You only need enough for the rest of the formation to draw supply, which is usually going to be 2-3% of your total tonnage. The majority of your infantry-based logistics should be held in the rear echelon and moved to the front automatically using the reinforcement/replenishment mechanics. The benefit of infantry-based logistics is that you get 25% more GSP per build point compared to LVH logistics, which is not negligible. 1000 GSP will cost you 2.48 BP for the LVH versus 2.0 BP for the INF. So it is true that having INF+LOG-S in your combat formations uses a little bit of tonnage, but the cheaper build cost is more than enough to offset this and with the BP you save you can just build more combat formations.
The benefit of LVH+LOG is that it is more tonnage efficient: 1000 GSP will take up 62 tons for the LVH versus 100 tons for the INF. This means LVH logistics are better in cases where tonnage is more important than build cost, which mostly means heavy armored offensive formations which are intended to make up the first drop wave of a major invasion force (when you are usually limited by transport capacity and need the most tonnage efficiency as a result). For defensive combat and smaller offensives where you are not limited by transport capacity, infantry-based logistics are usually going to be more efficient.
Also, note that boarding combat does not consume supplies, so you do not need logistics elements in a boarding combat formation.
For historical interest: It used to be that the benefit of LVH logistics was automatic resupply, whereas infantry-based logistics were MUCH more efficient but required manual replenishment during combat which was too much for most players. After the Unit Series and replenishment mechanics were added in 1.12, this was no longer the case and infantry-based logistics were strictly superior in every way, which is why the capacity of the vehicle-mounted LOG component was doubled to bring balance back to the Force, erm, I mean, the logistics.