Here's an expansion from 《史記》·《齊太公世家》that clarifies:
In this expansion, 實 means "truly"/"legitimately" (實) authorized to (得) launch expeditions (征) against them (之): launch expeditions against them with actual authority. So "legitimately" might be your best single word for the apparent function of 實 here.
實 as 的確 makes some sense in the expanded version, but still doesn't make sense as 的確 in the abbreviated form you cite in the 春秋左傳; so I would take that 實 as an abbreviated form of 實得, and gloss the original shorter phrase as "legitimately [authorized], go on the expedition" . It would be interesting to interpret 實 as "具備"(權力) but I can't find any clear evidence for that reading.
More examples of the 實得 usage would be useful: I'm having trouble finding other instances of this combination. It's worth noting that the above interpretation works quite well for the only other similar usage of 實 I can find, also from 《春秋左傳》·《昭公三十年》:「靈王之喪，我先君簡公在楚，我先大夫印段實往，...」.
Here are some additional examples of uses we can use to examine the problem. Based on searches from ctext.org.
This seems useful because the gloss as 確實 seems even more strained than the example you have provided. Here it seems to mean "May the Royal Protector [伯父] come here under authorization, so that We may congratulate him..." and then the Royal Protector comes in (I think - not sure what 侯氏 is, maybe one clan/surname/氏 of the 諸侯?).
I.e., the Royal Uncle [叔父, again, whatever] has dispatched 士季 to come under orders to renew old good will, in order to favor the Royal House.
There are other examples of "實來" in canon where it's not so clear it isn't simply meaning "really came". The advantage to the above selections, particularly the second, is that it's a real stretch to translate it as something like "X dispatched Y to actually visit..." - it just doesn't make very much sense.