Based on the your comments, I believe the issue lies with what you consider to be "carry." The reality is that there's no consensus. So let's take mini steps.
We'll start with what rates guys consider as "pure carry." In this most classical and fairly strict definition, carry is the deterministic component of expected returns – you know exactly what it is before you enter the trade. It is also very tangible, involving clear cash inflows/outflows. In this sense, @Daniel's answer is 100% correct as is: forwards, be it a forward bond position or even a forward starting swap, have no carry (no "pure carry" anyways).
There are several ways to think through this. First of all, recall that the futures price (excluding the embedded switch options, forward/futures difference, and other technicalities) is
$$\begin{align*} \text{Forward Price} &= \text{Spot Price} - (\text{Coupon Income} - \text{Financing Cost}) \\
&= \text{Spot Price} - \text{Pure Carry}.\end{align*} $$
This simple formula (and its equivalents) applies to all forward contracts. The futures price literally is the net result after the carry of the underlying has been removed.
Secondly, there is no tangible cashflows of any kind. By using a futures contract, you forgo coupon income from the underlying; nor do you pay a financing cost. Those have already been factored into the futures pricing.
Thirdly, as @Daniel has pointed out, carry basically provides you with a cushion – if carry is positive for a bond, yields can rise a little bit (by an amount equal to the difference between forward yield and spot yield) before you start losing money. For a futures contract, there's no such cushion at all. Yields start rising, you start losing money, because there's no coupon income to mitigate the capital losses.
This is not to say that there's no expected returns when you hold a futures contract and the world is static – now we're expanding the scope of the word carry. For clarify, I'll refer to this definition as "broad carry" – expected returns of an instrument when the world remains unchanged. When you allow for this broader definition, many things start to count.
For example, as you have pointed out, futures converge toward spot (by an amount equal to the underlying's pure carry). You can consider this to be a form of carry (I do!). Why isn't this "pure carry" though? Because it's neither tangible nor deterministic. It's not tangible because there are no real cashflows. It's not deterministic because bond futures allow for delivery one week after trading stops, so the classical futures/cash convergence may never happen.
Going further, bonds have expected rolldown returns that will flow through to futures – that could also count as a form of carry (some people do, others treat it as a separate concept). Bond futures also have an embedded delivery option, which can have time decay just like any other option. Bond futures may be mispriced relative to cash bonds, creating another source of convergence (toward fair value).
Anyways, I agree with @Daniel that strictly speaking, the carry of futures of zero. But if you're trying to think through what your expected returns might be if the world is unchanged, it's a much broader/messy definition. Depending how you trade and how you hedge, you are free to make discretionary decisions on what you want to count toward this "broad carry." It is probably best to list them out separately, so that you have a better idea about how reliable each component may be.
