I see some people say that using VPN can improve internet speed but I don't think that's true and in my experience most of the time using VPN makes my internet slower.
I think you may feel an improvement if your ISP uses a terrible DNS by default, or the websites you accessed have something against IPs from your nation, etc. But yeah in general the trade-off is speed, which shouldn't be that high if you use a decent VPN. At least that's how it is based on my experience, including accessing casino websites.
But I don't think they offer free service?
They do but they lack a lot of features[1]. It's probably worth as a trial to see if paying is worth it or not though, although you can't really tell if using different servers will affect your speed or not afaik.
And as I said before, the issue I am facing can be solved through several ways so if I should pay for vpn and still go with the other ways (which are more convenient) do you think that’s wise?
Probably not, especially if DNS is the culprit for your slow connection. But you can always do a trial and request a refund if it doesn't satisfy you. CMIIW.
[1]
https://protonvpn.com/free-vpn