Congratulations on receiving the offer!
To work in the UK, you can apply for the Skilled Worker visa (aka Tier 2) or Global Talent visa (aka Tier 1). The skilled worker visa need to be sponsored by your employer (the Uni) and you can't move jobs without re-applying for visa with another sponsor. The global talent visa allows you to work for any employer and switch jobs without changing your visa status.
Global Talent visa is more convenient, but also more expensive to apply for. The Skilled Worker visa is more restrictive, but less expensive for the applicant. However, the sponsor (University) will pay certain costs for sponsoring your application.
It is unusual for the Department Chair to have a say in which visa you should apply for. Frankly, it's not their remit. You can ask the University HR if the Uni sponsors Tier 2 visa. If the Chair insists you should apply for Global Talent, you can ask if the Department will reimburse your application fees. If it does not, then you should choose the option which better suits your needs and budget. Also, check carefully the eligibility conditions for both visas, as they are not the same! To apply for Tier 2 visa, you need a job offer from the University (which you have). But for Tier 1 visa you need a portfolio of your professional achievements, which need to be evaluated. The bar for getting the Tier 1 visa is higher, and generally speaking, your Department Chair should not have suggested that you apply for the visa which has a higher risk of being refused (unless they pay for it).
The healthcare surcharge is compulsory for both Global Talent and Skilled Worker visa. It makes you eligible for NHS treatment in the UK. You can't avoid paying this charge.
From your gross salary, the University will deduct taxes and national insurance. Both are compulsory and unavoidable (unless in very special cases). As a migrant, you are double-billed for NHS, in a form of the extra healthcare surcharge on top of your normal tax and national insurance contributions. However, this has nothing to do with your employer --- it's a recent development by the UK Government (welcome to the UK!)
Many Universities in the UK have an optional programs, which allow employees to sacrifice part of their salary for private healthcare, dental plans, private pension scheme, and other benefits. They are usually good and cost-effective options, which are worth considering. You can often pay for them by sacrificing your salary before tax, effectively getting a ~20% off. However, such initiatives can't be made compulsory. These are purely optional and you should be able to opt-out at any time.