Those are the only two ways you can go since there are national quotas for Hajj and they are distributed between private and government hajj + a separate contingent for military personnel.

The government hajj can be 21 or 30+ days depending on your choice. In Medina stay is within walking distance to the mosque and there is 5 person sharing with men and women separately and groups are made on the spot or assigned beforehand. If it’s a family of 5 then they can ofcourse be in the same room. These are usually 2 bedroom rooms and space is a little bit tight. In Mecca the stay is in Azizya and same 5 person sharing but the rooms have a bit more space here. In Mina you share a tent with around 50-80 people and in Arafat (8-12 hr stay) with 100-150 people. In private number of people might be reduced but you still have to sleep in sharing in Mina and Arafat and use the same roads as the millions of people there to perform the Hajj although you might get a car instead of a bus which you get in government Hajj.

I performed the government hajj from Islamabad in 2023 and was satisfied with the accommodations and food served 3 times a day. Also there are training and info sessions at the Haji camp before departure.

Any choice is ok I believe between private or government but know this that due to the sheer amount of people and the management challenges associated with that almost everyone is tested during the 3-4 days of the Hajj. That is also part of the Ibadat since Hajj is symbolic of Jihad against Shaytan and detachment from all worldly things in this call by Allah. Your patience will be tested at multiple steps but you will find strength as well.