I'm curious whether to see if routes have a "steam allowed" tick on them!
It's the other way round - Steam Locos are cleared for specific routes, rather than routes being cleared for Steam. This is because there can be quite wide ranging differences between classes of locos.
Since I'm sure many routes have never gotten steam since BR days, or at least have had years-long droughts in-between steam being used on railtours.
A fair amount of the network does get steam at least once a year, if you include positioning moves, although there will be many routes that don't because they're too busy or can't fit the locos down them.
This or vice versa, steam locomotives need certification to run on certain routes, something more specific than route availability
Steam is not issued permanent route clearance, only a 12-month temporary certificate. In nearly all respects the actual process for requesting and issuing clearance is the same as for any other train.
First the TOC needs to get the loco registered on the Mainline - this is usually for the duration of it's boiler certificate. This part of the process certifies the loco has compatible signalling, data recorders, radio, and other safety equipment. Once it's registered, the Operator submits a request to NR for clearance on specific routes. The gauging (always) and structure engineers (if needed) then assess the routes for size clearance and bridge strength, and will then return a Certificate of Temporary Gauging Authority that lists any restrictions or prohibitions (if required).
Once the CTGA is issued, it can be reused for any subsequent tours until it expires. Tours can run using the authority of multiple CTGAs if required (this is very common). CTGAs are typically returned with a specific loco number on them, but where locos of the same class share the same profile (e.g. the 2 SR Merchant Navy class locos) then CTGA issued for a different loco can be used - the profiles will have been verified as part of the mainline registering process. Black 5s could sometimes be cleared as the class rather than loco, as there's quite a few of them.
There were several locos that had a specific restriction on being only for the Whitby branch, although AIUI they still went through the above process.
Or is it just the railtour operator's decision on what to allocate?
So long as the operator allocated a loco that had all necessary clearances and could meet the timings of the tour, they were free to allocate any suitable loco on the day. Was quite common to be asked to time tours at 60mph, to allow flexibility in the case of a 75mph loco failing at short notice.