> Would the new amendment apply to the government, or to private sector, or to individuals?

It depends on how it's written, but there are no inherent limits here. You can make it do either.

> In any case, the 1st and the 28th? would continually be butting heads, since the 1st prohibits congress from passing laws that would enforce the 28th

Amendments, by definition, amend the constitution, as it was prior to them (meaning with all the previous amendments applied) so a later amendment can always override an older one explicitly, and even if it doesn't do so, the plain intent of the new amendment would likely suffice.