Everyone Active is a commercial company, so in principle these pools are little different from private club pools - they are just more accessible and affordable. And they do have a charitable arm. There are quite a few similar leisure ventre operators in the UK, and more recently quite a few public pools have been taken over by not-for-profit Trusts. Few pools are truly public facilities directly operated by local authorities any more.
France is now pretty much the same, with many municipal pools now run by private operators. However there are pools in most towns across France, just look up your nearest one! They usually have specific hours for lane swimming.
And there are 41 pools in Paris so you should find one to suit