It all depends on what you hunt, but the general answer (i.e. the one applicable in the most cases) is "no" solids are not the default way to go in a .375 H&H.
Big cats (leopard & lion): the consensus is on a rather soft expanding bullet. It needs to open immediately as there is much less meat and bones to go through. Personal choice: Nosler Partition (NP).
Buffalo: now that expanding bullets can be relied upon to
also penetrate deep, the consensus is on a tough expanding bullet. It needs to plow deep through a lot of meat and bones. There is too much risk of a solid going through and wounding a second animal in a herd when an A Frame or TSX does the job just as well, or actually better, and generally stays within. Personal choice: .470 NE because I like to hunt buff really up close and personal and I am an incurable hopeless romantic, but if I used a .375 it would be A Frame or hollow point controlled expansion mono-metal such as TSX.
Note: many PH / outfitters will not allow clients to use solids on buff anymore. They made sense when softs could not be relied upon to drive deep, but these times are over.
Heavies (elephant, hippo, OK, let's pretend: rhino): solids. An expanding bullet generally will not penetrate enough. The .375 will kill them very dead, but the .40+ and 45+ will hit them a lot harder, especially on body shots.
Tiny Tens: as stated by
Red Leg, solids, unless you want to shoot it, gut it, and quarter it all at once.
General plains game: obviously an expanding bullet (see "Weight").
Weight? I reckon that a 250 gr A Frame or TTSX that retains 90 to 95% weight penetrates with 225 to 240 gr. That is a lot more than the traditional cup & core bullets that made the 300 gr .375 legendary and that routinely lost 50+% and penetrated with only 150 gr. Simple physics. Personal choices: would be 300 gr solid on the heavies but again for me it is .470 territory; 300 gr NP for specialized lion or leopard hunt where you re-sight anyway; and 250, 260, 270 gr A Frame, TSX, TTSX, GMX, ETip (whatever tickles your fancy and/or shoots best in your rifle) as a do-it-all load for general bag buff & plains game. Lighter bullets (e.g. 235 or 250 gr) recoil less but keep in mind that even though they start faster they also have lower sectional density and ballistic coefficient... 260/270 gr might be the ideal compromise...
I would not carry 5 different loads in the field simultaneously - this is a recipe for disaster - but I would take different loads to Africa for different hunts (e.g. elephant vs. lion vs. general bag).