Before being thrown out of the TARDIS he makes the observation that it the Doctor who puts Clara in danger. He is the one that requires them to be rescued by people like Danny. This claim does seem to be justified.
The Doctor himself is aware that his companions are danger and if they travel with him long enough they will die. This was part of the reason that he tried to let Amy and Rory go, for fear of eventually standing over their graves (which is what happened).
This can be seen in ‘Deep Breath’ when the dinosaur that the Doctor whisked to Victorian London is incinerated. Unintentionally or not those who travel with the Doctor have a much reduced life expectancy.
This highlights the flaw in the Doctor travelling with only one companion. The very fact that he takes them to dangerous places invalidates any goodwill his rescuing them might have deserved because they wouldn’t need rescuing if not for him.
Within your own campaign having only two PCs, a Time Lord and his companion, leads to similar problems. The default assumption of the series is that the Time Lord will instigate the adventure by taking people to where the adventure is centred via his TARDIS. He could even provoke the danger into making its presence known.
The companion is expected to react, survive and if things get bad hope that the Time Lord can rescue them. Sometimes the Time Lord bites off more than they can chew and isn’t able to rescue their companion, either because they lack the skills or are physically detained elsewhere.
Having more than one companion solves this problem and indeed was the original setup. The 1st Doctor travelled with Susan, Ian and Barbara. When Susan left she was replaced by a number of different young women and once Ian and Barbara left Steve Taylor joined.
The 2nd Doctor travelled initially with Ben and Polly, with Jamie joining them later. When the two Londoners left they were replaced by Victoria and then Zoe. This ensured that aside for a few rare occasions he always travelled with at least two other companions.
The 3rd Doctor might be seen as the exception since he only had one companion (Liz Shaw, Jo Grant, Sarah Jane Smith) but their numbers were bolstered by the presence of the various UNIT personnel.
The 4th Doctor began travelling with two companions (Sarah Jane & Harry) before travelling mostly with Leela and Romana but the presence of K-9 similarly made them a trio.
By the end he was travelling with several companions who the 5th Doctor inherited. This incarnation continued to have crowded TARDIS, until the end when only Peri was left. The 6th incarnation continued to travel with only one companion (for the most part) until the 11th took both Amy and Rory with him.
The 11th Doctor era has several examples of how the trio worked well together. Indeed ‘The Power of Three’ bases a whole episode around the idea that the group of time travellers works very well together.
An argument could be made that Danny is presuming that Clara needs rescuing her and this could be seen to undermine her ability to make choices and look after herself. However within this triangle of relationships she doesn’t always have to accept the role of being the helpless one.
More than one person can ‘light the fire’ and more than one person can be the one to carry the others ‘out of fire’. This allows everyone to contribute and know that there is more than one person who can save them should they get into trouble.
Happily this setup should be easy to achieve in most roleplaying groups but in situations where there are only two players this can serve as a good reason why there should be an additional friendly NPC (typically the Time Lord) to be the instigator while the PCs can rescue each other.
Within the fiction of the show the Doctor has long worked out that he does need a companion. This revelation from Danny could prompt some self reflection and make him realise that one companion isn’t enough, he needs two or more.
There is evidence to suggest that his companion provides an alternative point of view that the Doctor needs. Following this reasoning having more opinions can only help the Doctor to see things from multiple points of view.
This could change the criteria that he usually uses to recruit companions. Typically he seems to look for very clever people, with an ability to think outside the box and have strong morals.
While he might certainly pick at least one companion for these qualities he might also look to take others with a range of different experiences. A historian might provide an ability to look at the bigger picture, as well as being familiar with any time period they travel to. A philosophy or religious expert could allow him to tap into a range of different way s to view the world while an activist could suggest new ways to do things.
These new companions need not agree with each. Indeed it might be beneficial for them to have contrary points of view. They should be unified on their purpose and the goals should align with the Doctor’s.
This could be akin to ‘Stark Trek’ where Kirk would consult with Spock and McCoy. The two Starfleet officers would often have opposing views but they respected their captain enough to let the final decision be his.
Some companions could be chosen for the very purpose of protecting the others (and the Doctor). The current incarnation of the Doctor might not like soldiers but he could see the wisdom of having people he can trust onboard who can ensure that no one dies because the Doctor acted without thinking.