I would say not "regardless of pitch". A lot theorise it rests the quicks, ties and end down, all manner of theories but not particularly considered in depth
Let's say the pitch is swinging, aussies are 100/4 on day one and suddenly England bring on the spinner to "rest the quicks", "tie and end down", spin it how you like (see what I did there
) Basically you have tied one hand around your balls, playing your least effective bowler in the conditions, based on not a lot
I don't like bowling attacks of five quicks, but rotated properly a four man attack should do the job if the conditions suit seam. If they do their job shouldn't be bowling long enough to need much rest, poor little lambs having to bowl 20-25 overs in 6+ hours of cricket. Playing a spinner then is preparing to bowl into day two, or putting it more cynically gifting the batting side however many runs he goes for
if you have a better than "reasonably good" spinner then no brainer, bound to play and take wickets in the 1st innings anyway, but how many are? Someone can fiddle with crocinfo if they like to whittle out spin conditions, but for most non-sub continent spinners the 1st innings will mostly be bowled in not the most helpful spin conditions
1st innings (of match, average for 2nd innings of match in brackets)
Ali : 31 wickets @ 54.23 (43.88)
Lyon : 125 wickets @ 31.87 (34.60)
A.Patel : 22 wickets @ 30.64 (28.42)
Leach : 20 wickets @ 46.55 (44.97)
Patel has better averages in innings 2 and 4 than 1 and 3, pretty good all round, but has played vast majority of his Tests in countries you'd expect to help spin (11/16)
could analyse more, would probably need to look at it in a better site than I am at the moment, largely as on laptop and not ideal, Of course there won't be a massive sample as you'd have to rule out the main bowlers of spin, India, given most of their 1st innings spin will be bowled in India. Pakistan's two spinners I looked at haven't really played enough to conclude much
of course how much a spinner bats can aid this, India in England prefer Jadeja over Ashwin but neither is useless with the bat. regardless of which innings Jadeja averages 43.48 with ball in England, Ashwin 28.11 including a 1st innings (of match) 4/62
isn't all about the 1st innings of the match of course, but in helpful seam conditions that can be the deciding factor, Ali has picked up wickets bowling last in England (and other innings granted), and got MOTMs, but seem to recall at least 1-2 matches where for instance Anderson and Broad, or just one or the other, did for the opponents in the 1st innings and Ali grabbed the headline performance bowling last
bottom line is whether or not you'll gain enough from the spinner in less than good conditions for bowling spin, it may not even get to the theoretical "spin wins the game bowling last on day five on a deteriorating pitch" scenario, games don't always go to days 4-5 PLUS, and this may be key, you can win Tests without spin playing a major role. so spin for me, picked "regardless", can just be a defensive approach hoping they'll play some part.
Helps if they can bat, a big part in why England kept picking Ali, can bat, bowls a bit of spin, seen as not costing too many games - and even though surprisingly Ali bowled the fewest overs of the four innings in the fourth innings, (423.3, 758, 498 and 422.1 in 1-4 respectively) he did take his wickets @ 23.17, was that worth it for the fewest overs (less than 1/5 by my calculation) and pretty poor returns in innings 1-3?
oh and one example of the free runs, in 1st innings 0/48 from Ali vs NZL, might otherwise have bowled them out for less than 350, kiwis racked up 454/8d in 2nd innings, Ali 1/73 so not quite "free" but 1/121 hardly causing serious damage, and England lost badly - Ali making 1 and 2 with the bat, wasn't quite the margin he conceded but didn't "tie up an end" and "keep the quicks fresh to run through the batting" as theory has it