does chess actually limit me to one lesson a week!

Yes and no.

If they gave it to you for free, you wouldn't think it was worth anything.

It looks like you have been a member for one month and have done 16 lessons.

That seems like more than one per week.

At any rate, why not do a trial premium membership?

You can do as many as you want.

#2 iirc lessons from very first courses seem to be not a subject to a limit (understandable – they teach basics of how pieces move, would be unfair to deny newcomers that), thus higher number. But after that they behave in typical free2play fashion and give just enough free content to players to get them interested but not enough to get satisfied. Don't excuse them, it's nothing more than a business (nothing actually evil tho). Trial premium goes there as well, existing to hook newcomers onto actually paying for more premium.