The basica problem with lack of unit AI

[Perturabo]

So, for example there's an infantry unit. When it and gains contact with enemy. Usually it suddenly turns out that it's outside cover. Even when there's a concrete building 20 metres away, it will stay there until ordered to move. So, it just stands in the open and takes casualties. The same with capturing a town, it's necessary to give orders to each squad (breaking up the unit) and wait up to several minutes for the order to get executed. I think that normally, these orders would be given by the platoon leader.

[Brahmin3]

Usually it suddenly turns out that it's outside cover.Even when there's a concrete building 20 metres away, it will stay there until ordered to move

Is that so unrealistic? Enemy can easily cut off the route to building by the heaviest concentration of fire, or surprised and frightened soldiers may be so pinned down as not having enough guts to make a dash through a bullet hailstorm.
In game terms, use smoke. Don't use big infantry formations for reconnaissance, it's scouts' job to "suddenly" discover a killing zone.

The same with capturing a town, it's necessary to give orders to each squad (breaking up the unit) and wait up to several minutes for the order to get executed.

Yes, as in real life, taking an urban area requires planning, and planning requires some time.
If you don't have time, you can order a quick assault with "use covered path" SOP. Soldiers will generally stick to the buildings in that case. Don't break up their path through urban are with a lot of waypoints - or they'll regroup between them out in the open.
Echeloned line formation can also help to squeeze a whole company inside an urban area. Many of them are just too small to accommodate the whole 1,5km-wide 12-squads US infantry line, for example.

[Perturabo]

Usually it suddenly turns out that it's outside cover.Even when there's a concrete building 20 metres away, it will stay there until ordered to move

Is that so unrealistic? Enemy can easily cut off the route to building by the heaviest concentration of fire, or surprised and frightened soldiers may be so pinned down as not having enough guts to make a dash through a bullet hailstorm.

It's unrealistic because they lack initiative and lack command on the level directly above them.

The same with capturing a town, it's necessary to give orders to each squad (breaking up the unit) and wait up to several minutes for the order to get executed.

Yes, as in real life, taking an urban area requires planning, and planning requires some time.
If you don't have time, you can order a quick assault with "use covered path" SOP. Soldiers will generally stick to the buildings in that case. Don't break up their path through urban are with a lot of waypoints - or they'll regroup between them out in the open.

They won't seek cover when reaching the final waypoint, though. The delay doesn't simulate planning - it simulates command and control delays - note how it increases with lack of HQ in range and increases even more with lack of radio. These delays happen because it is necessary to detach squads from platoon for repositioning of single squads - they are no longer under command of the platoon leader but under direct command of the player. And since they don't have radios it takes a lot of time for the orders to get through to them.
Often it's necessary to move a squad a square away to get to cover/get it into firing position but it takes 2 or more minutes because they have to get an order from Battalion/Brigade HQ not from the platoon leader. Once, I had to move a HMG team 40 metres forward because it didn't have LoS to enemy. It took 5 minutes for order to get through to it because it didn't have a radio. Also, order delays could use some consistency - some of them have immediate effect despite lack of link with HQ and stuff like that - for example using smoke, cancelling movement, etc.
The same with defence - it's necessary to detach every squad and move it to cover manually.

Two things could be done - a mode where the squads inside of a unit would move to cover and if a part of a unit is in a firefight, they would move to a place with los to enemy unit.

Then there's stuff like units not using bounding overwatch and similar small unit tactics. In Close Combat, it was necessary to use fire and movement inside squads, while currently in AB it's impossible to perform it on platoon-level and hard to perform it on company-level - as a result, attacking infantry often takes exceedingly high losses.

[Veitikka]

a mode where the squads inside of a unit would move to cover

When moving in a formation using the Covered path mode, the units could search the adjacent squares next to the waypoints, and move to the one with the best cover.

[Perturabo]

Sounds good.