Parts of the crew left the tank, but they had to abandon their assigned positions/guns in the tank to do that. There was no room to carry infantry sections and the crew consisted of a number of specialists: artillery gunners, MG gunners, loaders, driver, machinist(s), commander. All had infantry training ofc, but the tank was not designed to ferry "tankborne" inf troops, and each crew member manned a position - they didn't just sit inside and wait to dismount.
The A7V design was influenced by the british design and then even made bigger, with the goal to create a large platform that could basically serve as a rolling/armed fortress.
Initially, since the Germans were mainly on the defensive after their Verdun offensive, German departments and the leadership didn't really think that there was a need for a complex offensive platform/weapon. The British tanks could also be knocked out with field/infantry guns or even stalled/knocked out with bundled hand grenades, so - after the first shock - the German government thought that tanks would not be a successful platform, for quite a while.
This resulted in several departments only slowly starting to pay attention to the new type of vehicle/weapon. The departments' commitments were confused and partly even half-assed.
Still, the Verkehrstechnische Prüfungskommission (= traffic engineering assessment commission, short VPK) was pressured by the OHL (the Army's Supreme Command) to develop a tank in October 1916, where then the VPK explored possibilities with German corporations and car manfacturers. The authority over the coordination of those efforts happened to end up in the Abteilung 7. The department was then tasked with the construction of a tank under the code name A7V in November, 1916. The wooden 1:1 scale model was finished on January 16, so development/design had just taken ~8 weeks.
The designation "Verkehrswesen" (common translations and usages would be "traffic", "transportation", "traffic and communication" or "traffic system" and "transport system") did not mean that the subdivision no. 7 was a real/pure traffic or transportation department. One or another source claims that the department was founded to get the tanks built, where it then sounds like the "Verkehrswesen" designation was picked as code name for the tank development. On the other hand, Verkehrswesen also included all modern types of movement/ground vehicles or vessels, and the department was a subdivision of the war department, not a division of a civilian department.
Since the technology/the weapon was completely new to the german war admnistration, some subdivision of the war department had to be assigned to take care of the development.
In any case, you are drawing the wrong conclusion here.
The department then hired an engineer, Joseph Vollmer, who was an expert in car and truck development, and who had developed/invented the world's first truck-trailer combination ("road train") for AEG's car division NAG in 1903, the project was a cooperation of the body manufacturer Kühlstein and Vollmer, but who also developed a strong interest in tractor development.
Obviously, the engineer had the required expertise to develop large/heavy vehicles.
The fact that this particular department managed to come up with solid blueprints and with a 1st model quickly (some sources say that the first full-size wood model was developed/built within 6 weeks), does not mean that the vehicle was designed or meant to be a an APC or IFV, at all.
The experimental predecessor of the Mark I derived from a tractor. Technically, even the Mark I was still a tractor where then the engineers had built a crew compartment and gun sections around it.
The German engineer refined that concept and aimed to create a version that would offer more space for the crew (16 men), be able to support troops and knock out enemy tanks.
Since each the engineer and the Army insisted on a number of specifications, like the retractable commander compartment, which was demanded by the Army to allow relocation on trains (bridges/tunnels), and other complex features and revisions, the completion of an operational prototype was signific. delayed (until October 1917).
The development of the K-Wagen had even started before the A7V development. The K-Wagen was a real monster (120 tons, even 150 tons at one development stage), and it looked like a giant Mark I.
1 production version was finished and was almost delivered by the end of the war, with a 2nd vehicle finished but lacking the engines. After the armistice, the provisional German government then asked for a permit to
snaptube vidmate test drive the operational monster, but the Allies denied the request and ordered to scrap both vehicles.
One of the German light tank projects (the LK III, 1918) picked up the French approach, its predecessor LK II (1918) used parts of a civilian Daimler passenger car, and the first version LK I actually used the complete chassis of a civilian Daimler car. The tracks and some other features were rather copies of the successful British Whippet tank. All tanks were designed by Vollmer.
When the training of the crews for the first 5 A7V vehicles commenced (1918), coordination with infantry advances was part of the training.
The tank was designed to support the infantry but also to fight enemy tanks: In history's first tank battle (against british tanks) a tank of the 2nd AV7 Abteilung encountered 3 Mark IV tanks, 2 female versions and one male version, where it "knocked out" (heavily damaged) the 2 females but failed to attack/recognize the male Mark IV, which then hit the A7V so precisely, that the crew left the A7V. The male Mark IV was knocked out by direct German artillery fire eventually, iirc.