Many of the rules around hiring employees are national rather than EU, so anything that did the whole lot as a turnkey solution would probably have to vary significantly from one member state to another (and within the UK, which for now remains aligned with the EU rules as well).
You can certainly get software that handles things like payroll and statutory leave. Using it has become a practical necessity anyway since governments like the UK's started requiring payroll data to be filed electronically using approved software built on top of their API.
IMHO, the problem with becoming an employer for the first time is that there are so many things that an employer is now required to arrange that even if they are individually quite reasonable the cumulative effort can be a deterrent.
For example, if an employee at your tech firm uses a display screen, you must arrange and pay for an eyesight test with a suitably qualified professional if they ask for one. You must also pay for their glasses if they need an extra prescription specifically for display screen use, but you don't need to do this if their ordinary prescription is already suitable.
While you're dealing with H&S issues, don't forget to conduct your suitable and sufficient risk assessment for each new employee who is going to be working from home during the pandemic restrictions and then make any changes needed so they can do so in a safe and healthy way, whatever any of that actually means in practice.
The list of things you're automatically responsible for the moment you become someone's employer is pretty long in my country, and if you want to get it even close to right you'll probably end up dealing with lawyers, accountants, insurers, pension providers, HMRC, anyone who makes related software you now need to use, and from time to time possibly others like medical professionals.
As far as I'm aware, the closest thing to a turnkey solution that covers all of this is engaging an HR professional, presumably on a consultancy basis since you hardly need one full time if you're a bootstrapped startup looking for employee #1. But then that's more cost to deal with, and it's still going to take considerable time for one or more of your founders to make all the necessary arrangements even if you've brought in an expert to tell you what those arrangements need to be.