Your employer cannot write you a check, trusting you to put it in your HSA, and avoid any tax.
When your employer sets up an HSA for you, there is nothing special about that HSA that is any different than an HSA that you would set up yourself. Your employer could take the money that they deduct from your paycheck, add in any amount that they want to throw in, and send it directly to your own personal HSA that you set up yourself through your local bank. However, in my experience, employers generally don't do this; if the employer allowed it, then they would need to keep track of where everyone has their HSA and send lots of money to lots of different places, and also differentiate this from all the different bank accounts that they are direct depositing regular payroll.
Instead, it is easier for the employer to work with one bank/custodian to handle every employee's HSA contributions. Often, this is a firm that specializes in employee benefit services and is part of a package of services that they are handling for the employer.
If you don't like the HSA that your employer has set up for you, you can set up a second HSA that you like better. Just be aware that your annual HSA contribution limits are total; you can't double your HSA contribution limit by opening up another HSA. You are, however, allowed to transfer money from one HSA to another, so you could periodically move money from your employer's HSA to another one of your choosing. (There are specific rules about these transfers that need to be followed, of course.)
