The fix to the logistics system is extremely simple and could to some degree be implemented without really changing the current core systems.
1. Remove player requirements to place FOBs.
2. Remove squad size requirement to use a Logi.
3. Give Squad Leaders a shovel.
4. Equalize build radius and proximity radius on FOBs.
5. Change FOB Radio model into a Supply Dump model and remove ticket cost.
Aaaaaand that's it. You don't even really have to change the Radio model, it's all just a matter of changing a bunch of values in the current implementation and you'll have a logistics system that pretty much fixes all of the issues present currently. The reason you don't see dedicated logistics squads a lot right now is because a couple of direct and tangential game mechanics make them extremely difficult.
Squad is a 80 player game, meaning each team generally has a maximum of 40 players per team. The distribution of that manpower to various tasks, roles, and objectives is one of the most important parts of winning a match in Squad. The 3-man requirement to place a Radio forces any Logistics squad to have at least 3 players in their squad to be effective, however those 3 players have zero utilization for the majority of the match. The majority of the time, they do nothing other than sit in a truck. Basically, to operate a Logistics squad effectively, you have to disadvantage your team by locking up manpower in an idle situation for large portions of the match, and with just 40 players, having 5-10% of your team idle for most of the match is often a death sentence for the team. Even with 50 players, this equation won't change much.
In addition to being a monumental waste of manpower, playing second/third fiddle in a Logistics squad is BORING. You do spend the majority of your time sitting in a truck, looking out the window. And when you're not doing that, your face is likely in the ground watching a shoveling animation on repeat. While the lead has a lot of stuff to do, including driving, communicating with squads, planning, and so on, you have precisely nothing to do. You are literally only there to hit that 3-man requirement and to shovel.
If you try to operate a Logistics squad with less than 3 people, you have to depend on other squads to function. The problem is that they often don't have the time or ability to assist you. Most often, where other squads are is not where you need to build FOBs, and where you need to build FOBs is where nobody is, nobody can get to, and nobody wants to be. This results in situation where trying to organize anything with another squad will take 5-10 minutes of additional time. Finding a squad that has someone available, identifying the available squad members, picking up the available squad members, all of this takes time. Time that you cannot waste in a game where, despite its apparent "slow pace", seconds and minutes really do matter a lot. Again, this approach is often a death sentence for teams.
As a consequence of all this, emplacements have virtually no value, support FOBs are almost never made, and logistics becomes an intermittent need, resulting in squads just taking that 1500/1500 logi truck out to their objective, dropping down a FOB and HAB, and building a bunch of useless fortifications, leaving the logi truck abandoned there. With the aforementioned changes, the above issues would be resolved as the manpower requirement would drop, there would be no need for low-utility squad members, and logi squads could operate independently.
Emplacements and fortifications would suddenly have more value now since you can place them pretty much everywhere. There is often little value in having HMGs and TOWs at the locations where you have your spawn HABs, as they are generally located in areas with restricted vision, if it's not an on-objective FOB you probably don't want its location known, and you don't want to risk ten tickets placing an isolated FOB on a flanking hill just to have an HMG there. So the only places where emplacements and fortifications are useful to any substantial degree is exactly the places where the current game mechanics dictate that they should not be placed.
Take Kohat as an example. HMGs and TOWs on or around Mohd Zhai are virtually useless. They're either vision limited by surrounding buildings, trees, and tall grass fields, or they're on an exposed downward slope. Where you'd like to have the HMGs and TOWs is in the hills South of Mohd Zhai or further back in the plateau North-West of Mohd Zhai, where they can engage enemy vehicles overlooking the flag from the Eastern hills or engage infantry along the road, in the fields, or in the river East of Mohd Zhai. But you can't really place them there because the FOB costs 10 tickets if lost, so you either need a lot of manpower just to defend this one HMG FOB, or you leave it undefended as an easy target for roaming squads. It's a lose-lose situation. With the aforementioned changes, this and many more emplacement and fortification uses will suddenly be viable, opening up a whole new dimension of strategic and tactical possibilities.
The situation is the same with support FOBs, that is FOBs that operate as supply nodes or vehicle Repair FOBs. Because it is so hard to get anything built outside of where the squads are, meaning in the active combat areas, these are either never built or they are built in these active combat areas. This means that vehicles in need of repair and rearm generally only have the option of doing so either in the middle of the combat area or back at the main base. Again, with the aforementioned changes, vehicle squads would have no issue finding safe support FOBs outside of the combat area and far away from the main base. Squads and vehicles would have an easier time accessing supplies, as well.