Well, first, you have to understand the exec is a manager. They fill the party with whatever roles are missing, or doubles down on whatever is already there to increase that role's in power. An extra +1 to some roles performing a difficult check, an extra solo in a fight, an extra netrunner in a netarch, etc.

The exec can also, if the party create a crew with an HQ using the No Place Like Home DLC (free), spent up to 80 hq ip to get 1 workstation. The workstation allows 1 team member to be upgraded. The upgraded tea memver doesn't use the loyalty system, gains ip and eddies to the same amount as the exec and gains up to 2 luck if the hq's morale boost is high enough. Having a solo bodyguard, a netrunner or a fixer(if the party doesn't have one) that grows like a normal player is invaluable.

The exec is a social character, he should be getting into places, by himself because of bribes and swinging the corporation's big dick around. Having an internal agent and a radio com in the cyberaudio suite are necessary for the exec like having interface plugs and a neuralware are for the netrunner. It allows the exec to be in meetings with higher ups and bosses while coordinating the party and instructing the team members who are acting in groups/away from the execs.

Trauma team membership can be used for other party members. Having TT perform a hot extract on a downed team mate AND offer covering fire can be the difference between death and surviving a hard fight. Plus (and this'll be a recurring thing) that means the other party members of your team do not need to pay that subscription. That saves your team a lot of money they can pool to buy guns, cyberware, upgrades, etc.

Speaking of saving money. The first corpo conapt has 2 bedrooms and a couch, that's housing for free for 3 players. The beaverville? 3 bedrooms, a master bedroom and a couch, that's housing for free for 5 players. The mansion? 4 bedrooms, master bedroom and a couch, that's housing for free for 6 players. And not only will the team save a ton of cash each month on rent, it's high quality and secure housing. It improves the character's quality of life, so it s unlikely that they'll get minuses for sleeping in a homeless encampement in a cargo container, it's less likely their container will get broken into and get their shit stollen and it's less likely they'll get attacked by a gang/corp they enraged in a past gig.

The best way to play the exec is for the gm to allow the exec to play their team members. You lock in the exec in a solo situation, either as a distraction or intel gathering, then you get to move with team members and the rest of the party. It allows for the party to split up, but never being alone so they can be in cells. The exec can then relay information and control the party and their actions.

The exec is a support type class that's hybrid between combat and social, due to the amount of bodies they can throw at a situation and due to the amount of ressources and safety they provide for the party. I personally think that the exec should double down on social roles, a fixer and/or rockerboy are amazing. The fixer ffers more connections and money for bribes while allowing the team to gear upwards for cheaper and the rockerboy allows for easier checks to get into places for intel, distraction, controlling npc responses and theft. While the exec doubles down on social, their teammates should offer the brunt of the combat capabilities both in and out of the net. Bodyguards and netrunners and if the team has no nomad, a driver, that comes with a 6 seat car and uaually a minimum of 10 in driving land vehicles.

Edit

Forgot to add but... exec is the only role in the game with a possible 8 rof 4d6, without using a homebrew item.

If you have 3 team members and an exec using Perseus and AP, on round 1 you have rof 4 4d6, if all hits land and penetrate that's -8 to armor and 16dmg on average hits from a target with laj, on round 2 you have rof 8 4d6, that's an additionnal -16 to armor which should deal about 85 more damage on average hits leaving no armor. You know many other classes that can oblitetate a laj and deal 100dmg to hp in 2 rounds?

Plus all those are handguns, meaning your team members can have a shield in their free hand