This theory is mostly supported by the Doctors regenerations. The 1st, 5th, 6th, 9th, 10th and 11th all regenerate within the TARDIS itself. The 3rd Doctor was already dying from radiation sickness when he emerges from the TARDIS, suggesting that the process had already begun.
The 2nd Doctor regenerates off screen but does arrive in ‘Spearhead From Space’ apparently recently having changed (he is wearing his old clothes and collapses). It is also important to note that the Time Lords, despite having exiled him, leave him his TARDIS. This could indicate that they knew he would need it to regenerate if he were to be seriously injured and taking it away would be a death sentence.
The 4th Doctor regenerates outside of the TARDIS but only after the Watcher merges with him. It could be that this future version of the Doctor only appears during this regeneration specifically because the Doctor wasn’t in the TARDIS and so needed the ‘kickstart’ to begin the process. Once his companions are able to get the 5th Doctor into his ship the process can be completed.
The 7th Doctor regenerates in a morgue, leaving the 8th Doctor confused and unstable until he is able to return to the TARDIS. It is possible that if this didn’t happen his regeneration would have failed.
In ‘Turn Left’ Donna witnesses a reality in which the 10th Doctor drowns and doesn’t regenerate. The theory given is that the flood happened too fast for his body to regenerate but it could be that it was because he didn’t get to his TARDIS in time.
‘Human Nature’ establishes that the TARDIS has a Chameleon Arch that can change a Time Lord into a human. Since this obviously involves altering the subjects body it makes sense if this same technology is part of the regeneration process (and could mean that the TARDIS itself has some part to play in what the Doctor looks like and even what type of person he becomes).
There are exceptions, particularly with other Time Lords. In particular in ‘Last of the Time Lords’ the Doctor begs the Master to regenerate, without any concern for getting him into a TARDIS. Rather than the Master simply refusing to regenerate out of spite he physically couldn’t.
From the evidence provided in the series then there is an argument that Time Lord PCs can only regenerate if they are able to get into a TARDIS to start or stabilise the change. This makes their ship much more important.
If seriously injured they must get to the TARDIS. Story points can be spent to begin the transformation (regenerating as normal) but they will still need to get to their ship in order to stabilise. They should suffer physical and mental complications until they do.
Being unable to access the TARDIS isn’t just an inconvenience, it is potentially lethal. Not only can the Time Lord and his companions not leave but should he be injured there will be no chance of a successful regeneration.
Obtaining a TARDIS can therefore become important motivation for PCs and Time Lord NPCs stranded without one. Rogues and exiles would be haunted by the knowledge that while they have the potential to regenerate that they would still die without their own ship.
There are alternatives. It wouldn’t make sense if this were the only way for a Time Lord to regenerate as the impression given pre-Time War is that few actually had their own TARDIS or left the planet.
Rather than having to have one on stand-by for emergencies it makes more sense that the part of the TARDIS that is responsible for aiding regenerations could be built separately. This technology could then be kept in the home or in hospitals.
What this means post-Time War is that a Time Lord could build the vital, life-saving technology if they don’t have their own TARDIS. This should be a piece of portable technology (otherwise all Time Lords would carry it around with them) and should ideally be time consuming to build with some parts being hard to obtain.
This can give Time Lords characters a long term objective, especially those who are stranded. This makes it appropriate for a Time Lord PC in a UNIT campaign. It could also be the source of an adventure featuring a evil Time Lord, such as the Master, who might be obtaining parts for his own machine any way they can.
There are larger ramifications to this that will affect non-Time Lords. ‘Doctor Who The Movie’ shows the TARDIS bringing Grace and Chang Lee back to life, which could be part of being in temporal orbit but could equally be part of the same process that helps regeneration.
Potentially the same technology could be used to resurrect other characters. To restrict the use of this process the GM could require a Time Lord PC to sacrifice some of their regeneration energy (in much the same way that the Doctor heals River Song and she later saves him using the same process).
An adventure could be based around a race using a captured Time Lord to power their own Resurrection Machine. The PCs could rescue the Time Lord but could they accept the responsibility of bringing death back to the world?
Emphasising that a Time Lord can not regenerate on their own can make PCs far less blasé about dying. No longer can they sacrifice themselves without first thinking about whether they can get back to their TARDIS in time.