Benefit to disabling TLS1.1 and TLS1.2?

is there some good reason from a security perspective to allow only TLS 1.0, or is it simply pure "laziness"

It is in most cases just the TLS stack used. One of the most common stacks in web servers on UNIX/Linux is OpenSSL and the still widely used (and supported) versions 0.9.8 and 1.0.0 can do only TLS 1.0 and lower. TLS 1.1 and TLS 1.2 were only added with version 1.0.1 (04/2012). On the Windows side SecureChannel supports TLS 1.1 and TLS 1.2 since Windows 7 and Windows Server 2008 R2 (end of 2009).

And while changing the TLS stack might be doable for some sites it involves is most cases either the upgrade of the operating system or at least to run some parts of the system in a configuration not supported by the vendor.

and there is no good reason not to allow TLS1.1 and TLS1.2 in addition to TLS1.0?

If the server has support for TLS 1.2 there is no real reason to set the server up for only TLS 1.0. In fact, TLS 1.2 not only gives better security but also new cipher suites which can lessen the load on the server.

No, there is absolutely no security related reason to continue to support TLS 1.0, but there are several other business concerns which can twist the arm of a system engineer into allowing it. For larger sites, they may be trying not to leave people with older browsers out in the cold. For some situations, the person publishing the website needs to assume you are using I.E 6 with no updates.

It sounds ridiculous, and it is, but it's a fact of life.

That said, if the server doesn't support anything above TLS 1.0, that's a problem no matter how you slice it. You should e-mail them about it. Or, if it's not fixed, stop using the site.

Sometimes it only takes one angry e-mail to set things right.

Edit: I shall emphasize that supporting a legacy operating system or software stack is no excuse for exposing your users to encryption with major known faults.

There is no security benefit in disabling TLS1.1 and 1.2 in favor of 1. And there is actually no compatibility benefit either^*.

The reasons why some sites only support TLS1 is, that the software and/or configuration is simply too old.

Just stating the obvious, disabling all protocol versions older than TLSv1.2 is a security benefit as all of them have known weaknesses compared to 1.2.

^* On server side. It is unfortunately a bit different for clients which have to talk to very old and broken servers. They might not want to offer TLS1.1 or TLS1.2 for compatibility reasons. Or actually if they offer it, they might want to retry without.