LHR to Birmingham car service or train?

Take the train.

Coach to Reading Rail Station:

https://www.railair.com/

Then train to Birmingham

http://www.nationalrail.co.uk

These two websites will give you schedule and ticket information.

Or Elizabeth Line/tube to Marylebone then Chiltern Line to Birmingham Snow Hill or Birmingham Moor Street ( hourly)-https://timetables.chilternrailways.co.uk/#/timetables/1786/Table%202

Or Elizabeth Line/tube to London Euston then Avanti to Birmingham New Street- twice an hour- advance fares are around £20-£25 (this is the fastest route)
https://www.avantiwestcoast.co.uk/travel-information/plan-your-journey/timetables;
Or the slower London North Western Railway trains over almost the same tracks (but divert briefly via the Northampton Loop), advance fares as low as £8-
https://www.londonnorthwesternrailway.co.uk/sites/default/files/lnr_a_web_dec_2022_v1.pdf

On Chiltern fares as low as £10 can often be found, more commonly advance fares are around £20-£30.

There is nothing wrong with the Reading route on Cross Country trains (above) but will always be by far the most expensive- this is their timetable-https://www.crosscountrytrains.co.uk/media/6225/dec-22-book-1-final-web-version-v3_rakesh-bansal.pdf

reasonable summary above.

I've never heard of the Northampton line called a loop (usually a very short extra track often used for passing) despite having worked on it for many years.

The high single fare for the Railair coach plus the nearly always expensive Cross Country fares (needing multiple splits to get them half decent) sinks the that route.

One alternative not mentioned is Elizabeth Line to Hayes & Harlington, then go to Slough, change to GWR for Worcestershire Parkway, & there to Birmingham NS. Timewise it can be surprisingly similar to the slower via London options and sometimes cheaper all-in depending on the Advances cost. Will be less good after the May timetable though when the Worcester trains no longer stop at Slough.

You would not be allowed to travel from Heathrow to Birmingham via the Cross Country Reading route as that is not a permitted route (all fares are set by Avanti and by Chiltern)- according to the National Routeing Guide - which I have just looked up.
The Worcestershire Parkway route you would need 3 tickets- Heathrow to Hayes and Harlington (£7.20), an advance single Hayes and Harlington to Worcestershire Parkway which is about £15, then just buy a Cheap Day Single (no advance fares) at W Parkway to Birmingham New Street, which is £10.10- overall journey of around 3 hours, no real time saving, and no real cost saving either, But that's how you do it.
W Parkway being a very new station (and way oversized for it's traffic) it's a novel route.

There are all in through fares "Route Railair Reading" although these are more expensive than buying the coach separately so of academic interest only.

The Worcester route is a bit niche and I was more surprised that there was so little in it by time and cost than anything else. You could make it cheaper at present by taking a £2 bus ride to Slough instead!

Heathrow to Reading to Birmingham may be more expensive but it's the easiest. And after an all night flight, easy might be preferable.

Not everyone cares only about the cheapest. Sometimes easy takes a preference.

I was thinking of saving £5.40 by taking a London bus to Hayes and Harlington or West Drayton, as I'm not quite sure where route 8 is terminating in Slough due to the fire at Slough Bus Station, but the advance train fare from Slough to WP is £3 less than from H and H.

I did find a map somewhere of the temporary stops but can't trace it now.

Talking about easy there is also an hourly, at least, National Express Coach from Heathrow to Birmingham direct, some of which achieve a 3 hour or less timing, so as fast as any of the train routes.

The other issue with Reading to Birmingham is the packed out, short trains of Cross Country. You pay way over the odds for your ticket then end up on a rammed train.

Not so bad if it is starting from Reading.