You can request your own data here: https://employees.theworknumber.com/
You can generally do this for less than $100 per year in your home state. It can be a challenge to get the IRS to issue an EIN to such an entity, but if you read the rules you can find the exact flowchart of yes/no questions to force them to do it (pro-tip: do that first, so you actually create the business properly). As a bonus, you'll probably end up with a 20% discount on your income come tax time ([1][2]).
[1] https://www.irs.gov/pub/irs-utl/2018ntf-twenty-percent-small...
Part of your benefits as a W-2 employee don't really do you any good to have 2x of them. 401k plans - your max contribution is the total, not per employer. Health insurance? Well, maybe some people would be better to have two policies (but then you'll get demands for the rest of your life to document whether or not you have two policies for the purposes of coordination of benefits).
Honestly, having one W-2 and one or more 1099 job is better than one or more W-2 job - if you can get it that way.
The problem isn’t outsourcing a difficult and error-prone requirement that is never part of the core business (payroll and benefits). The problem is those outsourcing companies selling that data or using it for purposes other than managing payroll and benefits. Unfortunately we live in a world where anything we participate in, voluntarily or otherwise, potentially exposes us to data harvesting and our information sold to the highest bidder.