James Hamilton records his birthplace as Strathaven on the census records but I assume Strathaven covers both Strath and Avondale.
No. Other way round. Strathaven is a small town in the parish of Avondale, so (a) there are no separate parish records for Strathaven and (b) there will be records of people in Avondale who were not in Strathaven.
As for FamilySearch, if you find a listing there which does not match an actual record on Scotland's People, be very cautious. Basically, what is on FS is an
index to some (not all) of the original records, not actual records as such. So if something turns up on FS that isn't based on Scotland's People, you need to track down the source document to be sure that it's authentic and not a figment of someone's imagination.
Does anyone know if there are large chunks of the Strathaven registers missing
See
https://www.nrscotland.gov.uk/research/guides/old-parish-registers/list-of-old-parish-registers but make sure to look for Avondale, not Strathaven.
or is there anywhere other than Scotland's People I can search?
Not really. The vast majority of surviving Scottish church records are on Scotland's People. The only surviving ones missing are those of some minor denominations, including the Episcopal, Congregational and Baptist churches. Those are in various places including local authority, university or diocesan archives.
I also thought I had checked the 'other churches' search facility. It looks like this is the explanation that some births are on Scotland's People and some are not if the church was built in 1846. So, I take it the marriage in 1837/38 and earlier baptisms must have been elsewhere and most likely in a Catholic church.
Generally speaking, what determines whether or not they are on SP is whether or not the records, if they ever existed, have survived. The 'other churches' are mostly those whose registers are held by the National Records of Scotland rather than by the Registrar General. The date when the church was built is irrelevant because some congregations existed for a time before they could afford to build a church; and also many churches were built on the sites of, or as replacements for, older churches.
See the Statistical Account of Avondale
http://www.rootschat.com/links/01sx3/ for some information about which denominations existed in Avondale in the 1790s. There is no mention of Roman Catholics, Episcopalians, Congregationalists or Baptists. All the denominations actually named are splinter groups of the Church of Scotland.