I don't think this is possible, but can a WAP 1.x device view WAP 2 websites? Or even full HTML websites? My question specifically refers to older phones (e.g., http://www.gsmarena.com/ericsson_t20s-211.php) without better browsing capabilities than WAP 1.x.
Keep in mind this is WAP 1.x which isn't based on HTTP, as opposed to WAP 2.
The answer is not a clear yes or no, but rather that it depends upon the specific devices.
In the case of the Ericsson T20 that you mentioned, it is a NO. Looking at the WURFL database (http://wurfl.sourceforge.net/) indicates this device does not support XHTML, and that it actually has a preference for WML 1.1. This suggests that it does not even support the latest WAP 1.x standards (WAP June 2000 Release, commonly referred to as WAP 1.2.1, which uses WML 1.3 markup), but is an older WAP 1.1 device.
However, some WAP 1.x devices possibly could view WAP 2 websites, depending upon how the device developer implemented the browser. The only way to tell definitively is to look at the content types that the device indicates via HTTP headers.
A WAP 1.x device cannot communicate directly with a web server because it does not support a conventional TCP/IP stack, hence it must go via a WAP Gateway. But that does not preclude the WAP 1.x device from supporting more advanced markup formats such as WAP 2 (XHTML Mobile Profile + Wireless CSS), iMode (cHTML) or some subset of HTML etc.
One example of this is the Openwave v6 browser, commonly used in many WAP phones. This browser can support both WAP 1.x and WAP 2, and the markup formats indicated above. If the device developer decided not to use a TCP/IP stack (lower development effort, lower license costs, lower bandwidth required etc) then the device would be limited to using the WAP 1.x protocol stacks, and must go via a WAP Gateway to communicate with the web server. But it could still handle the more advanced markup formats.
The answer is simply no.
Take a look at the phone and look at the screen size the fact that it isn't even color how do you expect it to display any thing that is remotely Full HTML or WAP 2?
It's possible to display a converted form of the original page/site via WAP gateways which translate a server's HTTP/S communication to WAP 1 and conversely a phone's WAP 1 to HTTP/S: http://www.thewirelessfaq.com/how_does_a_wap_device_connect_to_the_internet
It's transparent to the users (aside from the pages being ugly), so it seems like they actually have an HTTP/S browser--when they in fact do not.